hcju/isabelleps huggingface dataset:
STEP_REF: fixes x :: "'b"   and x' :: "'a" assumes "(x, x') \<in> R"   and "b x"   and "b' x'"   and "I x"   and "I' x'" shows "f x \<le> \<Down> R (f' x')"
subtype_x_Sequence: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> Sequence \<sigma>" obtains \<tau>' :: "'a OCL_Types.type" where "\<tau> = Sequence \<tau>'" and "\<tau>' \<sqsubset> \<sigma>"
transfer: shows "(\<lambda>f g. if g = 0 then 0 else let n = ratfps_subdegree g; h = ratfps_shift n g in ratfps_shift n (f * inverse h)) = (/)"
leadsTo_Un: fixes F :: "'c program"   and A :: "'c set"   and C :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto> C"   and "F \<in> B \<longmapsto> C" shows "F \<in> A \<union> B \<longmapsto> C"
lstutter_extend: shows "lbisimulation (rel_of_pred rel_gc) (stutter_extend s1.system_automaton) (stutter_extend s2.system_automaton)"
less_not_refl2: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "m \<noteq> n"
residual_empty_bn_eq_iff: fixes \<alpha>1 :: "'a"   and P1 :: "'b"   and \<alpha>2 :: "'a"   and P2 :: "'b" assumes "bn \<alpha>1 = {}" shows "(\<langle>\<alpha>1,P1\<rangle> = \<langle>\<alpha>2,P2\<rangle>) = (\<alpha>1 = \<alpha>2 \<and> P1 = P2)"
nn_integral_le_const: fixes c :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and M :: "'c pmf" assumes "0 \<le> c"   and "AE x in measure_pmf M. f x \<le> c" shows "integral\<^sup>N (measure_pmf M) f \<le> c"
vbex_vifunion: fixes I :: "V"   and f :: "V \<Rightarrow> V"   and P :: "V \<Rightarrow> bool" shows "(\<exists>z\<in>elts (\<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)). P z) = (\<exists>x\<in>elts I. \<exists>z\<in>elts (f x). P z)"
cases: fixes a1 :: "'c \<Rightarrow> 'd"   and a2 :: "'c option"   and a3 :: "'c option"   and a4 :: "'d DBMEntry" assumes "dbm_entry_val a1 a2 a3 a4" obtains u :: "'c \<Rightarrow> 'd" and r :: "'c" and d :: "'d" where "a1 = u__" and "a2 = Some r__" and "a3 = None" and "a4 = Le d__" and "u__ r__ \<le> d__"   | u :: "'c \<Rightarrow> 'd" and c :: "'c" and d :: "'d" where "a1 = ua__" and "a2 = None" and "a3 = Some c__" and "a4 = Le da__" and "- ua__ c__ \<le> da__"   | u :: "'c \<Rightarrow> 'd" and r :: "'c" and d :: "'d" where "a1 = ub__" and "a2 = Some ra__" and "a3 = None" and "a4 = Lt db__" and "ub__ ra__ < db__"   | u :: "'c \<Rightarrow> 'd" and c :: "'c" and d :: "'d" where "a1 = uc__" and "a2 = None" and "a3 = Some ca__" and "a4 = Lt dc__" and "- uc__ ca__ < dc__"   | u :: "'c \<Rightarrow> 'd" and r :: "'c" and c :: "'c" and d :: "'d" where "a1 = ud__" and "a2 = Some rb__" and "a3 = Some cb__" and "a4 = Le dd__" and "ud__ rb__ - ud__ cb__ \<le> dd__"   | u :: "'c \<Rightarrow> 'd" and r :: "'c" and c :: "'c" and d :: "'d" where "a1 = u" and "a2 = Some r" and "a3 = Some c" and "a4 = Lt d" and "u r - u c < d"   | uu_ :: "'c \<Rightarrow> 'd" and uv_ :: "'c option" and uw_ :: "'c option" where "a1 = uu_" and "a2 = uv_" and "a3 = uw_" and "a4 = \<infinity>"
get_nth_i_def: fixes ba :: "'b list \<times> 'b list" shows "get_nth_i ba = hd (snd ba)"
prod_atLeastAtMost_code: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" shows "prod f {a..b} = fold_atLeastAtMost_nat (\<lambda>a. (*) (f a)) a b (1::'b)"
divide_less_eq_numeral1_1: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / numeral w < a) = (b < a * numeral w)"
mkVarChannelProc_pState_inv: fixes prog :: "program"   and p :: "pState"   and g :: "'a gState_scheme"   and v :: "procVarDecl" assumes "pState_inv prog p"   and "gState_inv prog g"   and "cl_inv (g, p)"   and "v \<in> statesDecls (states prog !! pState.idx p)" shows "pState_inv prog (snd (mkVarChannelProc v g p))"
scaler_column_fst_QR_decomposition: fixes j :: "'a"   and i :: "'a"   and A :: "((real, 'a) vec, 'b) vec" assumes "j < i"   and "rank A = ncols A" shows "column i (fst (QR_decomposition A)) \<bullet> column j A = 0"
cases: fixes a :: "'c"   and p :: "'c \<times> 'd" assumes "a \<in> Basic_BNFs.fsts p" obtains "a = fst p"
irreducible_abs_int_poly: fixes p :: "int poly" shows "irreducible (abs_int_poly p) = irreducible p"
simps_1: fixes NTDom :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NTCod :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'j"   and more :: "'m" shows "NTDom \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NTDom"
distinct_insort_on: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "distinct (insort_key_on r f x xs) = (x \<notin> set xs \<and> distinct xs)"
extract_var_monom_mult: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and m' :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and b :: "'b"   and v :: "nat" shows "extract_var (monom (m + m') (a * b)) v = extract_var (monom m a) v * extract_var (monom m' b) v"
FactRing_simps_1: fixes R :: "('e, 'f) ring_scheme"   and I :: "'e set" shows "R Quot I = \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, one = I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>, zero = I, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
sumlist_vec_index: fixes xs :: "'a vec list"   and n :: "nat"   and i :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> carrier_vec n"   and "i < n" shows "abelian_monoid.sumlist (module_vec TYPE('a) n) xs $ i = (\<Sum>x\<leftarrow>xs. x $ i)"
prems_1: shows "c \<in> C\<^bsub>\<V>\<^esub>"
ComSecTyping_single_is_sound: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool"   and DA :: "'b \<Rightarrow> 'e"   and lH :: "('e \<times> 'a \<times> nat) set"   and AssignSideCondition :: "'b \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> bool"   and WhileSideCondition :: "'a \<Rightarrow> bool"   and IfSideCondition :: "'a \<Rightarrow> ('a, 'b) MWLsCom \<Rightarrow> ('a, 'b) MWLsCom \<Rightarrow> bool"   and c :: "('a, 'b) MWLsCom" assumes "Type_System E BMap DA lH AssignSideCondition WhileSideCondition IfSideCondition"   and "Type_System.ComSecTyping AssignSideCondition WhileSideCondition IfSideCondition c"   and "unique_PPc c" shows "WHATWHERE.WHATWHERE_Secure (MWLs_semantics.MWLsSteps_det E BMap) E pp DA lH [c]"
l1: shows "\<forall>t i cid. trace init t final \<and> Marker \<in> set (msgs (S t i) cid) \<longrightarrow> (\<exists>j\<ge>i. Marker \<notin> set (msgs (S t j) cid))"
Eq\<^sub>3S: fixes v :: "i"   and F :: "\<Pi>\<^sub>3"   and G :: "\<Pi>\<^sub>3" shows "[F \<^bold>=\<^sub>3 G in v] = (F = G)"
composition_series_simple_group: shows "simple_group G = (\<GG> = [{\<one>\<^bsub>G\<^esub>}, carrier G])"
\<phi>_def: shows "\<phi> = (\<lambda>y x. y\<^sup>P \<^bold>= x \<^bold>& \<lbrace>x,H\<rbrace>)"
mkArr_Fun: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S f" shows "set_category_given_img.mkArr S img \<AA> (set_category_data.set S img (partial_magma.dom S f)) (set_category_data.set S img (partial_magma.cod S f)) (set_category_given_img.Fun S img f) = f"
inj_on_imageI: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e"   and A :: "'c set" assumes "inj_on (g \<circ> f) A" shows "inj_on g (f ` A)"
B: shows "\<forall>n. (a, b) \<in> F (Y n \<lbrakk>A\<rbrakk> S)"
counit_simps_4: shows "src \<epsilon> = trg (h \<star> f)"
WF: shows "wf_tr M (fixpt M (init_tr M))"
Suc_nat_number_of_add: fixes v :: "num"   and n :: "nat" shows "Suc (numeral v + n) = numeral (v + num.One) + n"
simps_5: fixes v :: "nat \<Rightarrow> 'a"   and l :: "linear_poly"   and r :: "rat" shows "v \<Turnstile>\<^sub>c EQ l r = (l \<lbrace> v \<rbrace> = r *R (1::'a))"
simps_1: fixes f :: "'a \<Rightarrow> 'a"   and p :: "'a"   and d :: "'a" shows "iter f 0 p d = d"
merge_fun: fixes rh :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "merge_on UNIV rh bo m" shows "merge_on UNIV ((\<circ>) rh) (rel_fun (=) bo) (merge_fun m)"
integrable_diff: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f integrable_on S"   and "g integrable_on S" shows "(\<lambda>x. f x - g x) integrable_on S"
properD1: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and h :: "heap"   and as :: "nat set" assumes "proper P"   and "P (h, as)" shows "in_range (h, as)"
ex_cong: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool" assumes "\<And>x. Q x \<Longrightarrow> P x = P' x" shows "(\<exists>x. Q x \<and> P x) = (\<exists>x. Q x \<and> P' x)"
finite_range_imageI: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'c" assumes "finite (range g)" shows "finite (range (\<lambda>x. f (g x)))"
E_full_prop: shows "edges (generate_valid_topology M (fully_connected G)) = E"
nub_nub_body_eq: shows "nub = fix\<cdot>nub_body"
ORRIGHT2_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set" shows "pi \<bullet> ORRIGHT2 (A OR B) X = ORRIGHT2 (A OR B) (pi \<bullet> X)"
irr: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and U :: "'c set" assumes "IrrFinGroupRepresentation G smult V"   and "RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V" shows "U = 0 \<or> U = V"
map_of_is_SomeI: fixes xys :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = \<lfloor>y\<rfloor>"
assms_3: fixes a :: "'b"   and b :: "'b" shows "f (a * b) = f a * f b"
safeAborts_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and s :: "('e, 'f, 'g) SLin_state" assumes "SLin \<delta> \<gamma> bot" shows "SLin.safeAborts \<delta> s \<equiv> if initialized s then SLin.initAborts \<delta> s else SLin.uninitAborts \<delta> s"
encR: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "encoding.indRelRPOp Enc (SourceTerm S) (TargetTerm (Enc S))"
emptyPre: shows "Lxy (take (index qs a) qs) S = []"
modulo_bit_def: fixes a :: "bit"   and b :: "bit" shows "a mod b = Abs_bit (Rep_bit a \<and> \<not> Rep_bit b)"
finite_set_choice: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite A"   and "\<forall>x\<in>A. \<exists>y. P x y" shows "\<exists>f. \<forall>x\<in>A. P x (f x)"
normalize_crossproduct: fixes q :: "int"   and s :: "int"   and p :: "int"   and r :: "int" assumes "q \<noteq> 0"   and "s \<noteq> 0"   and "Rat.normalize (p, q) = Rat.normalize (r, s)" shows "p * s = r * q"
allocationInjectionsUnivervseProperty: fixes G :: "'b set"   and N :: "'a set" shows "allAllocations N G = injectionsUniverse \<inter> {a. Domain a \<subseteq> N \<and> Range a \<in> all_partitions G}"
cases: fixes g :: "'g"   and a1 :: "'node list"   and a2 :: "'val \<Rightarrow> 'node option" assumes "g \<turnstile> a1\<Down>\<^sub>s a2" obtains "a1 = [Entry g]" and "a2 = ssaStep g (Entry g) 0 Map.empty"   | ns :: "'node list" and s :: "'val \<Rightarrow> 'node option" and m :: "'node" and i :: "nat" where "a1 = ns @ [m]" and "a2 = ssaStep g m i s" and "g \<turnstile> ns\<Down>\<^sub>s s" and "last ns = old.predecessors g m ! i" and "m \<in> set (\<alpha>n g)" and "i < length (old.predecessors g m)"
bounded_by_Cons: fixes xs :: "real list"   and vs :: "float interval option list"   and x :: "real"   and ivl :: "float interval" assumes "bounded_by xs vs"   and "x \<in>\<^sub>r ivl" shows "bounded_by (x # xs) (Some ivl # vs)"
UnderS_decr: fixes A :: "'a set"   and B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "A \<subseteq> B" shows "UnderS r B \<subseteq> UnderS r A"
subseqs_subset: fixes ys :: "'c list"   and xs :: "'c list" assumes "ys \<in> set (subseqs xs)" shows "set ys \<subseteq> set xs"
simps_6: fixes e :: "expr" shows "\<lbrace>Sqrt e\<rbrace> = (if \<lbrace>e\<rbrace> < 0 then 0 else sqrt \<lbrace>e\<rbrace>)"
lActive_active: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat"   and n :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<forall>n'>n. \<not> \<parallel>c\<parallel>\<^bsub>t n'\<^esub>" shows "\<parallel>c\<parallel>\<^bsub>t \<langle>c \<and> t\<rangle>\<^esub>"
adds_pp_def: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and t :: "'c"   and v :: "'b" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_pp pair_of_term t v = (t adds term_powerprod.pp_of_term pair_of_term v)"
mult_numeral_1_right: fixes a :: "'d" shows "a * Numeral1 = a"
eqv_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f \<sim> g) = (\<forall>x\<in>domain. \<forall>y\<in>domain. x \<sim> y \<longrightarrow> f x \<sim> g y)"
kstar_inv: fixes I :: "'b \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'b set" assumes "I \<le> |F] I" shows "I \<le> |F\<^sup>*] I"
Sum_bind_pmf: fixes Da :: "'f pmf"   and fa :: "'f \<Rightarrow> 'e pmf"   and Da' :: "'e pmf"   and Db :: "'h pmf"   and fb :: "'h \<Rightarrow> 'g pmf"   and Db' :: "'g pmf"   and p :: "real" assumes "Da \<bind> fa = Da'"   and "Db \<bind> fb = Db'" shows "Sum_pmf p Da Db \<bind> f_on2 fa fb = Sum_pmf p Da' Db'"
is_rewritable_alt_spp: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and u :: "'t" assumes "0 \<notin> set bs" shows "is_rewritable bs p u = is_rewritable_spp (map spp_of bs) (spp_of p) u"
bot_entails_all: fixes B :: "'f"   and N1 :: "'f set" assumes "B \<in> Bot" shows "{B} \<Turnstile> N1"
Inv_in_Hom_2: fixes t :: "'c local.term" assumes "Can t" shows "Inv t \<in> VHom (Cod t) (Dom t)"
Confidentiality_S: fixes A :: "agent"   and K' :: "nat"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and Ticket :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs"   and "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Key K \<notin> analz (knows Spy evs)"
simps_5: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_BisT tval aval indis (c1 | c2) = (PL_Indis.SC_BisT tval aval indis c1 \<and> PL_Indis.SC_BisT tval aval indis c2 \<or> PL_Indis.SC_discr aval indis (c1 | c2) \<or> PL_Indis.SC_WbisT tval aval indis (c1 | c2))"
t1_space_openin_delete_alt: fixes X :: "'b topology" shows "t1_space X = (\<forall>U x. openin X U \<longrightarrow> openin X (U - {x}))"
span_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "Modules.module scale" shows "(module.span scale S = module.span scale T) = (S \<subseteq> module.span scale T \<and> T \<subseteq> module.span scale S)"
NR\<^sub>N_bounds: fixes p :: "real"   and n :: "nat"   and u :: "nat"   and l :: "nat" shows "random_skip_list.NR\<^sub>N p n u l \<le> of_nat (l * n) + of_nat u * random_skip_list.NH\<^sub>N p n"
surjective: fixes r :: "('a, 'b, 'c, 'd, 'e, 'f) BEState_scheme" shows "r = \<lparr>es = es r, ps = ps r, pubActs = pubActs r, \<dots> = BEState.more r\<rparr>"
nummul_nb: fixes t :: "MIR.num"   and i :: "int" assumes "numbound0 t" shows "numbound0 (nummul t i)"
b: shows " t \<longrightarrow>\<^sub>1\<^sup>* t1"
class_of_0_eq: fixes U :: "'a set"   and U' :: "'a set" assumes "U \<in> {U. is_open U \<and> x \<in> U}"   and "U' \<in> {U. is_open U \<and> x \<in> U}" shows "stx.class_of U \<zero>\<^bsub>U\<^esub> = stx.class_of U' \<zero>\<^bsub>U'\<^esub>"
prems: shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b (r # sb)"
that_2: assumes "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) ((fst x, snd y) # zs)" shows "thesis"
eq_refl: fixes x :: "'b list"   and y :: "'b list" assumes "x = y" shows "prefix x y"
hull_rel_relcompI: fixes x :: "'b"   and y :: "'c"   and R :: "('b \<times> 'c) set"   and p :: "perm" assumes "(x, y) \<in> R" shows "(p \<bullet> x, y) \<in> hull_rel O R"
ldropn_Suc_conv_ldropn: fixes n :: "nat"   and xs :: "'c llist" assumes "enat n < llength xs" shows "LCons (lnth xs n) (ldropn (Suc n) xs) = ldropn n xs"
reds: shows "P \<turnstile> \<langle>e,(h, l(Vs [\<mapsto>] vs))\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"
insert_ident: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "(x::'b) \<notin> (A::'b set)"   and "(x::'b) \<notin> (B::'b set)" shows "((x::'b) \<triangleright> (A::'b set) = x \<triangleright> (B::'b set)) = (A = B)"
period_def: fixes x :: "'a" shows "period x = Inf {T \<in> existence_ivl0 x. 0 < T \<and> flow0 x T = x}"
interior_complement: fixes S :: "'a set" shows "interior (- S) = - closure S"
lformula0_decr0: fixes \<phi> :: "'a"   and k :: "'k"   and l :: "nat" assumes "lformula0 \<phi>" shows "lformula0 (decr0 k l \<phi>)"
refl_eq_ivl: fixes p :: "int extended \<times> int extended" shows "eq_ivl p p"
fresh_vquot_dbfm: fixes i :: "atom"   and V :: "name set"   and fm :: "dbfm" shows "i \<sharp> vquot_dbfm V fm = (i \<sharp> fm \<or> i \<notin> atom ` V)"
eval_monom_smallo': fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and e :: "real"   and x :: "real \<times> real list" assumes "basis_wf (b # basis)"   and "0 < e" shows "eval_monom x basis \<in> o(\<lambda>x. b x powr e)"
leg1_is_map: shows "is_left_adjoint f\<^sub>0gh\<^sub>1.p\<^sub>1"
simps_2: fixes r :: "'b rexp"   and c :: "'b"   and s :: "'b list" shows "slexer r (c # s) = (let (rs, fr) = simp (deriv c r) in case slexer rs s of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c (fr v)))"
orthogonal_clauses_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a"   and x :: "'a"   and c :: "real" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner a x" shows "real_inner.orthogonal inner a (scaleR c x)"
differentiable_rGamma_aux1: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and rGamma :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "class.Gamma dist uniformity open divide inverse times one plus zero minus uminus scaleR norm sgn rGamma"   and "\<And>n. z \<noteq> uminus (semiring_1.of_nat one plus zero n)" shows "let d = minus (THE d. LIM n sequentially. comm_monoid_add.sum plus zero (\<lambda>k. minus (inverse (semiring_1.of_nat one plus zero (Suc k))) (inverse (plus z (semiring_1.of_nat one plus zero k)))) {..<n} :> topological_space.nhds open d) (scaleR euler_mascheroni one) in LIM y topological_space.at_within open z UNIV. scaleR (inverse (norm (minus y z))) (plus (minus (rGamma y) (rGamma z)) (times (times (rGamma z) d) (minus y z))) :> topological_space.nhds open zero"
assms_2: shows "io_targets AB (qA, qB) w = {(pA, pB)}"
deriv_cdivide_right: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a"   and c :: "'a" assumes "f field_differentiable at z" shows "deriv (\<lambda>w. f w / c) z = deriv f z / c"
simps_24: fixes x61 :: "'c"   and x62 :: "('b, 'c) fm"   and x2 :: "'c" shows "(\<^bold>@ x61 x62) \<noteq> Nom x2"
T_AE_iterates: fixes P :: "(int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> bool" assumes "almost_everywhere MI P" shows "AE x in MI. \<forall>n. P ((TI ^^ n) x)"
ordertype_image_plus: fixes \<alpha> :: "V"   and u :: "V" assumes "Ord \<alpha>" shows "tp ((+) u ` elts \<alpha>) = \<alpha>"
comp_rtrancl_trancl: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set"   and s :: "'b"   and t :: "'b" assumes "R O S \<subseteq> S"   and "(s, t) \<in> (R \<union> S)\<^sup>* O S" shows "(s, t) \<in> S\<^sup>+"
length_list_update: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b" shows "\<^bold>|xs[i := x]\<^bold>| = \<^bold>|xs\<^bold>|"
linorder_ow_axioms: shows "linorder_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1"
singl_in_Bpow: fixes r :: "('a \<times> 'a) set"   and a :: "'b"   and A :: "'b set" assumes "Card_order r"   and "Field r \<noteq> {}"   and "a \<in> A" shows "{a} \<in> Bpow r A"
prod_state_def: fixes m :: "nat"   and u :: "complex Matrix.mat" shows "prod_state m u \<equiv> if state m u then \<exists>n p v w. state n v \<and> state p w \<and> n < m \<and> p < m \<and> u = v \<Otimes> w else undefined"
B0a: fixes x :: "i"   and y :: "i" shows "E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)"
rsp: shows "eq_onp overrider (\<lambda>x y. x) (\<lambda>x y. x)"
ins_invariant: fixes x :: "'a"   and S :: "'b" assumes "is_node x"   and "invariant S"   and "\<not> memb x S" shows "invariant (ins x S)"
ap_pmf_def: fixes f :: "('e \<Rightarrow> 'd) pmf"   and x :: "'e pmf" shows "f \<diamondop> x = map_pmf (\<lambda>(f, x). f x) (pair_pmf f x)"
Lfilled_imp_exists_Lfilled_exact: fixes n :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and LI :: "e list" assumes "Lfilled n lholed es LI" shows "\<exists>lholed' ves es_c. const_list ves \<and> Lfilled_exact n lholed' (ves @ es @ es_c) LI"
is_first_cons: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" shows "is_first x (y # ys) = (x = y)"
intTF: fixes \<sigma> :: "'a"   and a :: "'b" shows "intTF \<sigma> a"
to_rat_real_alg_def: fixes x :: "real_alg" shows "to_rat_real_alg x = (case to_rat_real_alg_main x of None \<Rightarrow> 0 | Some q \<Rightarrow> q)"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "\<chi>og.map f = C.null"
assms_2: shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>)\<cdot>r' \<noteq> \<bottom>"
bhv2cnf_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n \<equiv> \<langle>c \<and> t\<rangle> + (n - (the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1))"
distribute_fixpoint_over_setmonotone_comp: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set"   and I :: "'a set" assumes "setmonotone f"   and "setmonotone g"   and "(f \<circ> g) I = I" shows "f I = I \<and> g I = I"
hoare3o_valid_def: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>3\<^sub>' {P} c { Q}) = (\<exists>k>0. \<forall>ps n. P (ps, n) \<longrightarrow> (\<exists>ps' ps'' m e e'. (c, ps) \<Rightarrow>\<^sub>A m \<Down> ps' + ps'' \<and> ps' ## ps'' \<and> k * n = k * e + e' + m \<and> Q (ps', e)))"