hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'a OCL_Types.type"   and a2 :: "'a OCL_Types.type" assumes "to_ordered_collection a1 a2" obtains \<tau> :: "'a OCL_Types.type" where "a1 = Collection \<tau>__" and "a2 = Sequence \<tau>__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Set \<tau>'__" and "a2 = OrderedSet \<tau>'__"   | \<tau> :: "'a OCL_Types.type" where "a1 = OrderedSet \<tau>''__" and "a2 = OrderedSet \<tau>''__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Bag \<tau>'''__" and "a2 = Sequence \<tau>'''__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Sequence \<tau>" and "a2 = Sequence \<tau>"
inject_1: fixes x2 :: "complex mat"   and y2 :: "complex mat" shows "(Utrans x2 = Utrans y2) = (x2 = y2)"
imethds_norec: fixes G :: "prog"   and md :: "qtname"   and i :: "iface"   and sig :: "sig"   and mh :: "mhead" assumes "iface G md = Some i"   and "ws_prog G"   and "table_of (imethods i) sig = Some mh" shows "(md, mh) \<in> DeclConcepts.imethds G md sig"
simps_3: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Mult a b) vs = interpret_floatarith a vs * interpret_floatarith b vs"
gr_implies_not_zero: fixes m :: "'b"   and n :: "'b" assumes "m < n" shows "n \<noteq> (0::'b)"
atLeastLessThanPlusOne_atLeastAtMost_int: fixes l :: "int"   and u :: "int" shows "{l..<u + 1} = {l..u}"
in_bitset_bound: fixes b :: "nat"   and n :: "nat" assumes "b \<in> bitset n" shows "2 ^ b \<le> n"
nsubst_ode: fixes I :: "('sf, 'sc, 'sz) interp"   and ODE :: "('sf + 'b, 'sz) ODE"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "osafe ODE"   and "OadmitFO \<sigma> ODE U"   and "\<And>i. dsafe (\<sigma> i)" shows "ODE_sem I (OsubstFO ODE \<sigma>) (fst \<nu>) = ODE_sem (adjointFO I \<sigma> \<nu>) ODE (fst \<nu>)"
full_rank_imp_is_basis: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "rank A = ncols A" shows "is_basis (columns A) \<and> card (columns A) = ncols A"
append_xml': fixes ind :: "char list"   and i :: "nat"   and xml :: "xml"   and r :: "char list"   and s :: "char list" shows "shows_XML_indent ind i xml (r @ s) = shows_XML_indent ind i xml r @ s"
J: shows "g (Suc (nat (padic_val p f + padic_val p g))) = j * p ^ nat (padic_val p g) \<and> \<not> p dvd int (nat j)"
L_transform_formula_def: shows "L_transform_formula \<equiv> (FL_Formula.formula.Rep_formula ---> id) L_transform_Tree\<^sub>\<alpha>"
P: shows "P = (\<lambda>((s, l, r), n). s = Suc (2 * n))"
qItem_simps_10: fixes delta :: "'e"   and qinpa :: "'c \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm option"   and qbinpa :: "'d \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qAbs option" shows "qGood (qOp delta qinpa qbinpa) = (qGoodInp qinpa \<and> qGoodBinp qbinpa)"
word_le_def: fixes a :: "'k word"   and b :: "'k word" shows "(a \<le> b) = (uint a \<le> uint b)"
extra_axiom_def: shows "extra_axiom \<equiv> SOME extra_axiom. \<forall>e. eval_fm e extra_axiom"
Nsetn_sub_mem: fixes l :: "nat"   and n :: "nat" assumes "l \<in> {j. j \<le> n}" shows "l \<in> {j. j \<le> Suc n}"
try_spmf_cong: fixes p :: "'b spmf"   and p' :: "'b spmf"   and q :: "'b spmf"   and q' :: "'b spmf" assumes "p = p'"   and "\<not> lossless_spmf p' \<Longrightarrow> q = q'" shows "TRY p ELSE q = TRY p' ELSE q'"
map_FGco_def: fixes l1 :: "'aj \<Rightarrow> 'aw"   and co1 :: "'ak \<Rightarrow> 'ax"   and co2 :: "'am \<Rightarrow> 'az"   and co3 :: "'an \<Rightarrow> 'ba"   and co4 :: "'ao \<Rightarrow> 'bb"   and co5 :: "'al \<Rightarrow> 'ay"   and co6 :: "'as \<Rightarrow> 'be"   and contra1 :: "'bc \<Rightarrow> 'ap"   and contra2 :: "'bd \<Rightarrow> 'aq"   and contra3 :: "'bf \<Rightarrow> 'at"   and contra4 :: "'bg \<Rightarrow> 'au" shows "map_FGco l1 co1 co2 co3 co4 co5 co6 contra1 contra2 contra3 contra4 = map_F l1 co1 co5 (map_G co1 co2 co3 co4 contra1 contra2) co3 co6 contra1 contra3 contra4"
rfloor_def: fixes x :: "real" shows "rfloor x = real_of_int \<lfloor>x\<rfloor>"
A: shows "trnr\<^sub>\<eta> v \<in> hom (v \<star> g) u \<rightarrow> hom v (u \<star> f)"
gallery_least_length: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "chamber D"   and "C \<noteq> D" shows "gallery (C # (ARG_MIN length Cs. gallery (C # Cs @ [D])) @ [D])"
prems_2: shows "head \<notin> fst ` set (xs @ [x])"
join_top: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<squnion> \<top> = \<top>"
category_axioms: shows "category (\<cdot>\<^sub>A\<^sub>2\<^sub>x\<^sub>A\<^sub>1)"
sumeq: shows "(h\<^sub>0' ++ h\<^sub>1) a = (h\<^sub>0 ++ h\<^sub>1) a"
IH: assumes "safe s r"   and "g \<in> isin s r" shows "owns s r = \<lfloor>g\<rfloor>"
setcompr_eq_image: fixes f :: "'c::type \<Rightarrow> 'b::type"   and P :: "'c::type \<Rightarrow> bool" shows "{(f::'c::type \<Rightarrow> 'b::type) x |x::'c::type. (P::'c::type \<Rightarrow> bool) x} = f ` {x::'c::type. P x}"
W1: shows "D = T \<longrightarrow> A B CTSP Q R"
const_def: fixes x :: "'b"   and uu :: "'c" shows "const x uu = x"
simps: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'d bset"   and a2 :: "'e bset" shows "rel_bset R a1 a2 = (\<exists>x. a1 = map_bset fst x \<and> a2 = map_bset snd x \<and> set_bset x \<subseteq> {(x, y). R x y})"
inflstsI2: fixes a :: "'b"   and A :: "'b set"   and t :: "'b llist" assumes "a \<in> A"   and "t \<in> A\<^sup>\<omega>" shows "a ## t \<in> A\<^sup>\<omega>"
simple_linux_router_def: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and mlf :: "'b word \<Rightarrow> 48 word option"   and ifl :: "interface list"   and p :: "'b Simple_Packet.simple_packet_ext" shows "simple_linux_router rt fw mlf ifl p \<equiv> iface_packet_check ifl p \<bind> (\<lambda>_. let rd = routing_table_semantics rt (p_dst p); p = p\<lparr>p_oiface := output_iface rd\<rparr>; fd = simple_fw fw p in (case fd of Decision FinalAllow \<Rightarrow> Some () | Decision FinalDeny \<Rightarrow> None) \<bind> (\<lambda>_. let nh = fromMaybe (p_dst p) (next_hop rd) in mlf nh \<bind> (\<lambda>ma. Some (p\<lparr>p_l2dst := ma\<rparr>))))"
spmf_of_set_def: fixes A :: "'c set" shows "spmf_of_set A = (if finite' A then spmf_of_pmf (pmf_of_set A) else return_pmf None)"
sentence_starts_with_nonterminal: fixes N :: "symbol"   and u :: "symbol list" assumes "is_nonterminal N"   and "derives u []" shows "\<exists>X r. u @ [N] = X # r \<and> is_nonterminal X"
assms_5: shows "2 ^ card vs - 1 < length (as_proj as vs)"
rel_fun_ball: fixes I :: "'a set"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f' :: "'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> rel_fun R (=) (f i) (f' i)" shows "rel_fun R (=) (\<lambda>x. \<forall>i\<in>I. f i x) (\<lambda>y. \<forall>i\<in>I. f' i y)"
t: shows "rawpsubstT n (zip (map Var us) (map snd txs)) = n"
Cons_eq_tl_iff: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(x # xs = tl ys) = (\<exists>y. ys = y # x # xs)"
hidden_last_index: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "hidden (xs @ [x]) (last_index xs x)"
divide_right_mono_neg: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "c \<le> (0::'a)" shows "b / c \<le> a / c"
refl: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l, Z\<rangle>"
assms_3: shows "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
thetaSeqW_Wbis: shows "thetaSeqW \<subseteq> Wbis"
hn_refine_frame: fixes P' :: "assn"   and c :: "'c Heap"   and Q' :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and m :: "'d nres"   and P :: "assn"   and F :: "assn" assumes "hn_refine P' c Q' R m"   and "P \<Longrightarrow>\<^sub>t F * P'" shows "hn_refine P c (F * Q') R m"
assms_2: shows "y1 \<sqsubseteq> y2"
VLambda_vempty: fixes f :: "V \<Rightarrow> V" shows "VLambda 0 f = 0"
theI2: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
diff: fixes V :: "'c set"   and f :: "'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "linearform V f"   and "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "f (x - y) = f x - f y"
partitionI: fixes B :: "'d set set"   and A :: "'d set" assumes "\<Union> B = A"   and "\<And>b1 b2. \<lbrakk>b1 \<in> B; b2 \<in> B; b1 \<inter> b2 \<noteq> {}\<rbrakk> \<Longrightarrow> b1 = b2" shows "Congruence.partition A B"
compactin_imp_closedin: fixes X :: "'a topology"   and S :: "'a set" assumes "Hausdorff_space X"   and "compactin X S" shows "closedin X S"
cases: fixes x :: "'b rule list" obtains "x = []"   | uu_ :: "action" and rs :: "'b rule list" where "x = Rule (MatchNot MatchAny) uu_ # rsa__"   | vb :: "'b" and va :: "action" and rs :: "'b rule list" where "x = Rule (Match vb__) va__ # rsb__"   | vc :: "'b" and va :: "action" and rs :: "'b rule list" where "x = Rule (MatchNot (Match vc__)) vaa__ # rsc__"   | vc :: "'b match_expr" and va :: "action" and rs :: "'b rule list" where "x = Rule (MatchNot (MatchNot vca__)) vab__ # rsd__"   | vc :: "'b match_expr" and vd :: "'b match_expr" and va :: "action" and rs :: "'b rule list" where "x = Rule (MatchNot (MatchAnd vcb__ vd)) vac__ # rse__"   | vb :: "'b match_expr" and vc :: "'b match_expr" and va :: "action" and rs :: "'b rule list" where "x = Rule (MatchAnd vb vc) vad__ # rsf__"   | va :: "action" and rs :: "'b rule list" where "x = Rule MatchAny va # rs"
elims_3: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real"   and l0 :: "real"   and x :: "real \<times> real" assumes "Table f1 f2 f1' f2' e c l0"   and "\<not> Table.invar f1 f2 c l0 x" obtains n :: "real" and l :: "real" where "x = (n, l)" and "\<not> (l0 \<le> l \<and> (l0 \<le> l / c \<longrightarrow> f1 * l \<le> n) \<and> n \<le> f2 * l)"
term_ok_fv_imp_typ_ok_pre: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and x :: "variable"   and \<tau> :: "typ" assumes "is_std_sig \<Sigma>"   and "term_ok' \<Sigma> t"   and "(x, \<tau>) \<in> fv t" shows "typ_ok_sig \<Sigma> \<tau>"
simps_3: fixes ts :: "ct list"   and t1 :: "t"   and t2 :: "t" shows "select_return_top ts (TSome t1) (TSome t2) = (if t1 = t2 then TopType (take (length ts - 3) ts @ [TSome t1]) else Bot)"
map_transfer: fixes Rb :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and Sd :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "((Rb ===> Sd) ===> rel_fset Rb ===> rel_fset Sd) (|`|) (|`|)"
open_vimage_snd: fixes S :: "'b set" assumes "open S" shows "open (snd -` S)"
f_independent: shows "\<forall>a m. a \<in> F \<longrightarrow> f (g m) a = f m a"
a_minus_def: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
heap_node: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" shows "Leftist_Heap.linorder_class.heap (node l a r) = (Leftist_Heap.linorder_class.heap l \<and> Leftist_Heap.linorder_class.heap r \<and> (\<forall>x\<in>Tree2.set_tree l \<union> Tree2.set_tree r. a \<le> x))"
no_head_closed: fixes C :: "'b list set"   and b :: "'b list" shows "\<langle>no_head C b\<rangle> = no_head C b"
prems_1: fixes q :: "'a"   and m :: "nat" assumes "poly (pCons c p) (Fraction_Field.Fract a b) = Fraction_Field.Fract q (b ^ m)" shows "thesis"
Pd'_tax_range: fixes d :: "'b" shows "Field (Pd'_tax d) \<subseteq> {x. Xd x = d}"
preserves_dom: fixes f :: "'a" assumes "\<tau>.A.arr f" shows "\<tau>.B.dom (\<tau>'.map f) = G (\<tau>.A.dom f)"
wpt_choice: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.wpt one times inf top (inf x y) = inf (mbt_algebra.wpt one times inf top x) (mbt_algebra.wpt one times inf top y)"
tt_ge_iff: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(v \<preceq> punit.tt p) = ((p \<noteq> 0 \<or> v = punit.min_term) \<and> (\<forall>u. u \<prec> v \<longrightarrow> lookup p u = (0::'c)))"
I_minWeightBasis_final: fixes T :: "'a set" assumes "I_minWeightBasis (T, {})" shows "minBasis T"
wt_elim_cases_24: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and c1 :: "stmt"   and c2 :: "stmt"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (If(e) c1 Else c2)\<Colon>x" obtains "x = Inl (PrimT Void)" and "E,dt\<Turnstile>e\<Colon>-PrimT Boolean" and "E,dt\<Turnstile>c1\<Colon>\<surd>" and "E,dt\<Turnstile>c2\<Colon>\<surd>"
matrix_matrix_mult_inner_mult: fixes A :: "((real, 'c) vec, 'b) vec"   and B :: "((real, 'a) vec, 'c) vec"   and i :: "'b"   and j :: "'a" shows "(A ** B) $ i $ j = row i A \<bullet> column j B"
DBuffer_def: shows "DBuffer \<equiv> Init DBInit \<and> \<box>[DBNext]_(inp, mid, out, q1, q2) \<and> WF(DBDeq)_(inp, mid, out, q1, q2) \<and> WF(DBPass)_(inp, mid, out, q1, q2)"
intros_3: fixes a :: "'c"   and b :: "'c" assumes "a < b" shows "Le a \<prec> Le b"
UP_subring_deriv: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S" shows "deriv g a = derivative (R\<lparr>carrier := S\<rparr>) g a"
interleave_cases: fixes l :: "'b list"   and l1 :: "'b list"   and l2 :: "'b list" assumes "l \<in> l1 \<otimes> l2" obtains "l = []" and "l1 = []" and "l2 = []"   | a :: "'b" and l' :: "'b list" and l1' :: "'b list" where "l = a__ # l'__" and "l1 = a__ # l1'" and "l'__ \<in> l1' \<otimes> l2"   | a :: "'b" and l' :: "'b list" and l2' :: "'b list" where "l = a # l'" and "l2 = a # l2'" and "l' \<in> l1 \<otimes> l2'"
scalar_left_one: fixes nn :: "nat"   and v :: "'a list"   and i :: "nat" assumes "vec nn v"   and "i < nn" shows "scalar_prod (vec1 nn i) v = v ! i"
components_are_iso: fixes a :: "'c" assumes "C.ide a" shows "D.iso (G a)"
opp_fold_A_chamber_images_2: shows "opp_fold_A ` C = D"
neg_one_in_nonpos_Ints: shows "- (1::'a) \<in> \<int>\<^sub>\<le>\<^sub>0"
ordered_fmap_image: fixes S :: "('a \<times> 'b) fset"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "is_fmap S" shows "map (\<lambda>(a, b). (a, f a b)) (ordered_fmap S) = ordered_fmap ((\<lambda>(a, b). (a, f a b)) |`| S)"
simps_6: fixes m :: "nat"   and r :: "mregex" shows "ok m (MStar r) = ok m r"
sin_cos_between_zero_two_pi: fixes x :: "real" assumes "0 < x"   and "x < 2 * pi" shows "sin x \<noteq> 0 \<or> cos x \<noteq> 1"
fds_abs_converges_mult_const_right: fixes f :: "'b fds"   and s :: "'b"   and c :: "'b" assumes "fds_abs_converges f s" shows "fds_abs_converges (f * fds_const c) s"
Max_wrt_preorder: shows "Max_wrt R = {x \<in> alts. \<forall>y\<in>alts. R x y \<longrightarrow> R y x}"
pdevs_domain_def: fixes x :: "'d pdevs" shows "pdevs_domain x = {i. pdevs_apply x i \<noteq> (0::'d)}"
apath_append_iff: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list"   and v :: "'a" shows "apath u (p @ q) v = (apath u p (awlast u p) \<and> apath (awlast u p) q v \<and> set (awalk_verts u p) \<inter> set (tl (awalk_verts (awlast u p) q)) = {})"
red_rtrancl_diff_induct: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c"   and P :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool" assumes "(red F)\<^sup>*\<^sup>* (p - q) r"   and "P p p"   and "\<And>y z. \<lbrakk>(red F)\<^sup>*\<^sup>* (p - q) z; red F z y; P p (q + z)\<rbrakk> \<Longrightarrow> P p (q + y)" shows "P p (q + r)"
inter_overappr_def: fixes X :: "'e set"   and Y :: "'e set" shows "inter_overappr X Y = SPEC (\<lambda>R. X \<inter> Y \<subseteq> R \<and> R \<subseteq> X)"
fps_monom_poly_aux: fixes c :: "'b"   and k :: "nat" shows "fps_monom_poly_aux c k = smult c (eulerian_poly k \<circ>\<^sub>p [:0::'b, c:])"
eqOn_FV_sem_NE: fixes \<phi> :: "'aprop dfmla"   and env :: "pvar \<Rightarrow> nat"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and env1 :: "pvar \<Rightarrow> nat"   and \<pi>l1 :: "('state \<times> 'aprop set) stream list" assumes "cpt (FV \<phi>) env \<pi>l"   and "cpt (FV \<phi>) env1 \<pi>l1"   and "local.eqOn (FV \<phi>) env \<pi>l env1 \<pi>l1"   and "\<pi>l \<noteq> []"   and "\<pi>l1 \<noteq> []"   and "last \<pi>l = last \<pi>l1" shows "sem \<phi> env \<pi>l = sem \<phi> env1 \<pi>l1"
closedin_self: fixes S :: "'a set" shows "closedin (top_of_set S) S"
un: shows "poly_cond (abs_int_poly (reflect_poly p))"
characteristic_weak_formula_is_weak: fixes P :: "'state" shows "weak_formula (characteristic_weak_formula P)"
minus_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "x - y = (fst x - fst y, snd x - snd y)"
aborting_consistent: shows "\<forall>s t u. vpeq u s t \<longrightarrow> aborting s u = aborting t u"
Max_wrt_among_map_relation_vimage: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'a set" shows "f -` Max_wrt_among le A \<subseteq> Max_wrt_among (map_relation f le) (f -` A)"
between_in: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a"   and x :: "'a" assumes "before vs ram1 ram2"   and "pre_between vs ram1 ram2"   and "x \<in> set vs" shows "x = ram1 \<or> x \<in> set (between vs ram1 ram2) \<or> x = ram2 \<or> x \<in> set (between vs ram2 ram1)"
mono_SucI1: fixes X :: "nat \<Rightarrow> 'b" assumes "\<forall>n. X n \<le> X (Suc n)" shows "monoseq X"
Maschke: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "\<exists>W. (RModule FG (\<cdot>) W \<and> W \<subseteq> V) \<and> V = W \<oplus> U"
that: fixes \<C>' :: "'a set set" assumes "S \<subseteq> \<Union> \<C>'"   and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"   and "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> \<C>'. X \<inter> V \<noteq> {}}" shows "thesis"
p_def: fixes s :: "'a"   and s' :: "'a"   and t :: "real" shows "p s s' t = \<P>(\<omega> in K.lim_stream (0, s). trace_in {s'} t s \<omega>)"
cut_less_Max_less: fixes I :: "'b set"   and t :: "'b" assumes "finite (I \<down>< t)"   and "I \<down>< t \<noteq> {}" shows "Max (I \<down>< t) < t"
representative_of_pd_valuation: fixes P :: "('a \<Rightarrow> ant) set" assumes "P \<in> Pds" shows "valuation K (\<nu>\<^bsub>K P\<^esub>)"
H_fcs_eq_for_a: fixes b :: "'b" assumes "b \<in> arcs H \<inter> G.face_cycle_set a" shows "H.face_cycle_set b = S"
zeta_Re_ge_1_nonzero: fixes s :: "complex" assumes "1 \<le> Re s"   and "s \<noteq> 1" shows "zeta s \<noteq> 0"
set_add_atoms: fixes r :: "'b rexp"   and as :: "'b list" shows "set (add_atoms r as) = atoms r \<union> set as"
indep_subset: fixes X :: "'a set"   and Y :: "'a set" assumes "indep_in (insert x X) X"   and "Y \<subseteq> X" shows "indep_in (insert x X) Y"
arc_injective: fixes x :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])" shows "injective x"
swap_msgs_Trans_Recv: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
ltl_prop_equiv_def: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> = \<A> \<Turnstile>\<^sub>P \<psi>"
antisym: fixes b :: "'b list"   and a :: "'b list" assumes "prefix b a"   and "prefix a b" shows "a = b"
e_typing_imp_list_types_agree: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and vs :: "v list"   and ts' :: "t list"   and ts :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> $$* vs : ts' _> ts' @ ts" shows "list_all2 types_agree ts vs"
ide_some_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "ide (some_src \<mu>)"
p_fun_zero: fixes x :: "'a"   and y :: "'a" assumes "is_p_fun x" shows "x ; y \<cdot> x ; - y = 0"
op_in_ops: shows "op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
sv_TS_defs_3: shows "sv_trans = (\<Union>r S v. \<Union> (range (sv_round r S v)))\<^sup>="
eq_refl: fixes x :: "'a word"   and y :: "'a word" assumes "x = y" shows "x \<le>s y"
interval_lebesgue_integrable_bigo_left: fixes f :: "real \<Rightarrow> real"   and A :: "real"   and g :: "real \<Rightarrow> real"   and B :: "real" assumes "f \<in> O[at_right A](g)"   and "continuous_on {A<..B} f"   and "set_borel_measurable borel {A<..<B} f"   and "interval_lebesgue_integrable lborel (ereal A) (ereal B) g"   and "A < B" shows "interval_lebesgue_integrable lborel (ereal A) (ereal B) f"
lossless_restrict_gpvI: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "lossless_gpv \<I> (restrict_gpv \<I> gpv)"
countable_UN_J: shows "countable (\<Union> (range J))"
jumpF_eq: shows "jumpF_pathfinish (part_circlepath z r st tt) z0 = jumpF (\<lambda>i. f i / g i) (at_left tt)"
ivl_disj_un_6: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{l..<u} \<union> {u} = {l..u}"
rep_eq: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b multiset" shows "count (filter_mset x xa) = (\<lambda>x. if x x then count xa x else 0)"
mndet_unit: fixes a :: "'d"   and P :: "'d \<Rightarrow> 'd process" shows "mndet {a} P = (a \<rightarrow> P a)"
termMOD_igWlsAbsIsInBar: shows "igWlsAbsIsInBar termMOD"
permutations_not_nil: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "permutations xs = concat (map (\<lambda>x. map ((#) x) (permutations (remove1 x xs))) xs)"
awp_subset: fixes F :: "'b program"   and A :: "'b set" shows "awp F A \<subseteq> A"
simps: fixes vs :: "nat list"   and f :: "facetype" shows "(Face vs f)\<^bsup>op\<^esup> = Face (rev vs) f"
Skolem_list_nth: fixes k :: "nat"   and P :: "nat \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<forall>i<k. \<exists>x. P i x) = (\<exists>xs. length xs = k \<and> (\<forall>i<k. P i (xs ! i)))"
elims: fixes x :: "('a, 'b, 'c, 'd, 'e, 'f) paramS"   and xa :: "'c \<times> 'f"   and y :: "('a, 'b, 'c, 'd, 'e) qAbs set set" assumes "absOfS x xa = y" obtains uu_ :: "'c \<Rightarrow> 'd list" and uv_ :: "'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set list" and ALF :: "'c \<times> 'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs set list" and uw_ :: "('c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set option) list" and xs :: "'c" and s :: "'f" where "x = ParS uu_ uv_ ALF uw_" and "xa = (xs, s)" and "y = set (ALF (xs, s))"
minus_less_iff_ennreal: fixes b :: "ennreal"   and a :: "ennreal"   and c :: "ennreal" assumes "b < \<top>"   and "b \<le> a" shows "(a - b < c) = (a < c + b)"
vmatch_vmatch'_eq: fixes p :: "pat"   and v :: "value" shows "vmatch p v = ??.Value.vmatch' p v"
nonzero: shows "coeffs p ! n \<noteq> (0::'a)"
ord_eq_less_trans: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a = b"   and "b <s c" shows "a <s c"
ordRels_def_1: shows "ordLeq = {(r, r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"