hcju/isabelleps huggingface dataset:
inter_rel_br: fixes a :: "'d \<Rightarrow> 'f set"   and I :: "'d \<Rightarrow> bool"   and b :: "'e \<Rightarrow> 'f set"   and J :: "'e \<Rightarrow> bool" shows "\<langle>br a I, br b J\<rangle>inter_rel = br (\<lambda>(x, y). a x \<inter> b y) (\<lambda>x. I (fst x) \<and> J (snd x))"
image_mset_const_eq: fixes c :: "'a"   and M :: "'b multiset" shows "{#c. a \<in># M#} = replicate_mset (size M) c"
nth_default_Nil: fixes dflt :: "'f"   and n :: "nat" shows "nth_default dflt [] n = dflt"
abs_eq: fixes x :: "nat" shows "Var x = Abs_linear_poly (fun_var x)"
surjective: fixes r :: "'a point2d_scheme" shows "r = \<lparr>xc = xc r, yc = yc r, \<dots> = point2d.more r\<rparr>"
no_subcls1_Object: fixes D :: "qtname"   and G :: "prog"   and P :: "bool" assumes "G\<turnstile>Object\<prec>\<^sub>C1D" shows "P"
less_diff_eq: fixes 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 a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (minus c b) = less (plus a b) c"
double_SN: fixes a :: "lam"   and b :: "lam"   and P :: "lam \<Rightarrow> lam \<Rightarrow> bool" assumes "SN a"   and "SN b"   and "\<And>x z. \<lbrakk>\<And>y.  x \<longrightarrow>\<^sub>\<beta> y \<Longrightarrow> P y z; \<And>u.  z \<longrightarrow>\<^sub>\<beta> u \<Longrightarrow> P x u\<rbrakk> \<Longrightarrow> P x z" shows "P a b"
tendsto_enn2ereal_iff: fixes f :: "'b \<Rightarrow> ennreal"   and x :: "ennreal"   and F :: "'b filter" shows "((\<lambda>i. enn2ereal (f i)) \<longlongrightarrow> enn2ereal x) F = (f \<longlongrightarrow> x) F"
is_volatile_simps_2: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_volatile (Read\<^sub>s\<^sub>b volatile a t v) = volatile"
Infinitesimal_HFinite_scaleHR: fixes x :: "real star"   and y :: "'a star" assumes "x \<in> Infinitesimal"   and "y \<in> HFinite" shows "scaleHR x y \<in> Infinitesimal"
mat_col_first_last_append: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat" assumes "dim_col A = m + n" shows "mat_col_first A m @\<^sub>c mat_col_last A n = A"
affine_imp_convex: fixes s :: "'a set" assumes "affine s" shows "convex s"
nempty_input: fixes nid :: "'nid"   and t :: "'a \<Rightarrow> cnf"   and n :: "'a" assumes "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>"   and "honest nid" shows "pin (\<sigma>\<^bsub>nid\<^esub>t n) \<noteq> {}"
case_5: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x51 :: "nat"   and x52 :: "'b"   and x53 :: "'b" shows "(case RecvMarker x51 x52 x53 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f5 x51 x52 x53"
lv_evt_defs_2: fixes mru_votes :: "process \<Rightarrow> (nat \<times> val) option"   and Q :: "process set"   and v :: "val" shows "opt_mru_guard mru_votes Q v \<equiv> Q \<in> Quorum \<and> (let mru = opt_mru_vote mru_votes Q in mru = None \<or> (\<exists>r. mru = Some (r, v)))"
symm_closure_of_symm_rel: fixes Rel :: "('c \<times> 'c) set" assumes "sym Rel" shows "symcl Rel = Rel"
narrow2_acom: fixes C2 :: "'a acom"   and C1 :: "'a acom" assumes "C2 \<le> C1" shows "C1 \<triangle> C2 \<le> C1"
at_within_eq_at_right: shows "at 0 within {0..} = at_right 0"
countable_cases_card_of: fixes A :: "'a set" assumes "countable A" obtains "finite A" and "|A| <o |UNIV|"   | "infinite A" and "|A| =o |UNIV|"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre (C\<^sub>1;; C\<^sub>2) Q = pre C\<^sub>1 (pre C\<^sub>2 Q)"
lset_P_V: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "lset P \<subseteq> V\<^bsub>G\<^esub>"
iso_char_1: fixes \<mu> :: "('a E.term, 'a) arr" shows "local.iso \<mu> = (arr \<mu> \<and> B.iso (Map \<mu>))"
uint_take_bit_eq: fixes n :: "nat"   and w :: "'b word" shows "uint (take_bit n w) = take_bit n (uint w)"
coll_self: fixes z :: "real \<times> real" shows "coll 0 z z"
tMap_tLast_inv: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace"   and f' :: "'c \<Rightarrow> 'a"   and t' :: "'c Trace" assumes "tMap f t = tMap f' t'" shows "f (tLast t) = f' (tLast t')"
pred_fmap_fmpred: fixes P :: "'b \<Rightarrow> bool" shows "pred_fmap P = fmpred (\<lambda>_. P)"
sets_embed_measure': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj_on f (space M)" shows "sets (embed_measure M f) = {f ` A |A. A \<in> sets M}"
Nmlize_Trg_1: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Nmlize V (bicategorical_language.Trg trg t) = bicategorical_language.Trg trg (bicategorical_language.Nmlize V t)"
list_seq_length_mono_weak: fixes seq :: "nat \<Rightarrow> 'b list"   and i :: "nat"   and j :: "nat" assumes "\<And>i. \<exists>x. seq (Suc i) = x # seq i"   and "i \<le> j" shows "length (seq i) \<le> length (seq j)"
prems_3: shows "\<forall>c. c \<in> set (clause # Wl') \<or> c \<in> set newWl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
WTrtFAcc: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and U :: "ty"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "P,E,h \<turnstile> e : U"   and "class_type_of' U = \<lfloor>C\<rfloor>"   and "P \<turnstile> C has F:T (fm) in D" shows "P,E,h \<turnstile> e\<bullet>F{D} : T"
cod_char: fixes f :: "'a" shows "S.cod f = (if S.arr f then B.cod f else B.null)"
Range_mono: fixes r :: "('e \<times> 'f) set"   and s :: "('e \<times> 'f) set" assumes "r \<subseteq> s" shows "Range r \<subseteq> Range s"
return_def: fixes a :: "'c" shows "IO.return a \<equiv> Abs_io (Pair a)"
simps: fixes m :: "int"   and a :: "int"   and n :: "nat"   and p :: "int \<times> int \<times> int" shows "fast_ppoint_mult m a n p = (if n = 0 then (0, 0, 0) else if n mod 2 = 0 then mpdouble m a (fast_ppoint_mult m a (n div 2) p) else mpadd m a p (mpdouble m a (fast_ppoint_mult m a (n div 2) p)))"
prems_1: shows "mtype(CT,m,D) = Cs \<rightarrow> C0"
simps_1: fixes \<phi>\<^sub>1 :: "nat formula"   and \<phi>\<^sub>2 :: "nat formula" shows "disj_to_dimacs (\<phi>\<^sub>1 \<^bold>\<or> \<phi>\<^sub>2) = disj_to_dimacs \<phi>\<^sub>1 @ disj_to_dimacs \<phi>\<^sub>2"
Says_K5: fixes servK :: "nat"   and A :: "agent"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (knows Spy evs)"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Says A B \<lbrace>servTicket, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"
nb: fixes v :: "'a" assumes "v \<in> {v |v. v \<in> set (\<Pi>\<^sub>\<V>) \<and> (\<Pi>\<^sub>G) v \<noteq> None}" shows "cnf (encode_state_variable t (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>G) v)) = {{literal_formula_to_literal (encode_state_variable t (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>G) v))}}"
PUT: fixes x :: "Var"   and f :: "Field"   and e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "CONTEXT x = low"   and "GAMMA f = low"   and "Expr_low e" shows "G \<rhd> Put x f e : Sec (\<lambda>(s, t, \<beta>). \<exists>r l C Flds. fst r x = RVal (Loc l) \<and> r \<equiv>\<^sub>\<beta> t \<and> lookup (snd r) l = Some (C, Flds) \<and> s = (fst r, (l, C, (f, evalE e (fst r)) # Flds) # snd r))"
transfer_tau: fixes s :: "'s net_state"   and s' :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state" assumes "(s, \<tau>, s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n" obtains \<sigma>'' :: "nat \<Rightarrow> 'g" and \<zeta>'' :: "'l net_state" where "((\<sigma>, \<zeta>), \<tau>, \<sigma>'', \<zeta>'') \<in> automaton.trans (opnet onp n)" and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>'' j = \<sigma> j" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>'', \<zeta>'')"
distinct_1: fixes x21 :: "'e"   and x22 :: "'e list" shows "[] \<noteq> x21 # x22"
i_Exec_Comp_Stream_Acc_LocalState_def: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c \<equiv> (localState \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) \<div>\<^sub>l k"
thetaSeqZO_ZOretr: shows "thetaSeqZO \<subseteq> ZOretr (thetaSeqZO \<union> ZObis)"
assms_1: shows "freshEnv ys y (rho [x \<leftarrow> X]_xs)"
fair_aux: fixes s :: "nat \<Rightarrow> nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and i :: "nat"   and j :: "nat"   and k :: "nat"   and t :: "nat" assumes "A.is_run s"   and "i < j"   and "cc (s i) \<le> k"   and "k < cc (s j)"   and "t < N"   and "tts (s i) t = WAIT k" shows "\<exists>l\<ge>i. l < j \<and> tts (s l) t = HOLD k"
qAFreshAll_qSwapAll_id: fixes zs :: "'a"   and z1 :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and z2 :: "'b"   and A :: "('c, 'd, 'a, 'b, 'e) qAbs" shows "(qAFresh zs z1 X \<and> qAFresh zs z2 X \<longrightarrow> X #[[z1 \<and> z2]]_zs = X) \<and> (qAFreshAbs zs z1 A \<and> qAFreshAbs zs z2 A \<longrightarrow> A $[[z1 \<and> z2]]_zs = A)"
simps_5: fixes f1 :: "'c btree \<Rightarrow> 'b"   and f2 :: "'c btree \<Rightarrow> 'c \<Rightarrow> 'c btree \<Rightarrow> 'b"   and x1 :: "'c btree" shows "(case T\<^sub>i x1 of T\<^sub>i x \<Rightarrow> f1 x | Up\<^sub>i x xa xb \<Rightarrow> f2 x xa xb) = f1 x1"
assms_2: shows "G \<subseteq> \<langle>C\<rangle>"
UP_ring_def: fixes R :: "('c, 'd) ring_scheme" shows "UP_ring R \<equiv> ring R"
lub_in_lattice: fixes cl :: "'b potype"   and S :: "'b set" assumes "CL cl"   and "S \<subseteq> pset cl" shows "lub S cl \<in> pset cl"
space_closed: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "subset_class \<Omega> M" shows "M \<subseteq> Pow \<Omega>"
pull_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "pull w1 w2" shows "kauff_mat w1 = kauff_mat w2"
external_equiv_internal_invariant_vimage: fixes P :: "'a set"   and S :: "('a, 'b) spec"   and Q :: "'b set" assumes "P = obs S -` Q" shows "(oreach S \<subseteq> Q) = (reach S \<subseteq> P)"
refl_le: shows "refl {(x, y). x \<le> y}"
infnorm_2: fixes x :: "(real, 2) vec" shows "infnorm x = max \<bar>x $ 1\<bar> \<bar>x $ 2\<bar>"
span_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "Modules.module scale" shows "Modules.module.span scale S = module.subspace scale hull S"
op_type_det: fixes op :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and k :: "call_kind"   and \<tau> :: "'a OCL_Types.type"   and \<pi> :: "'a OCL_Types.type list"   and \<sigma> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type" assumes "op_type op k \<tau> \<pi> \<sigma>"   and "op_type op k \<tau> \<pi> \<rho>" shows "\<sigma> = \<rho>"
z: shows "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p"
lborel_integral_real_affine: fixes c :: "real"   and f :: "real \<Rightarrow> 'a"   and t :: "real" assumes "c \<noteq> 0" shows "integral\<^sup>L lborel f = \<bar>c\<bar> *\<^sub>R (LBINT x. f (t + c * x))"
wf_trancl: fixes r :: "('f \<times> 'f) set" assumes "wf r" shows "wf (r\<^sup>+)"
scaleC_prod_def: fixes r :: "complex"   and A :: "'d \<times> 'e" shows "r *\<^sub>C A = (r *\<^sub>C fst A, r *\<^sub>C snd A)"
le_norm_blinfun: fixes f :: "'d \<Rightarrow>\<^sub>L 'c"   and x :: "'d" shows "norm (blinfun_apply f x) / norm x \<le> norm f"
final_i_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event llist" assumes "cocomputation channel trans send recv init"   and "distributed_system.cotrace channel trans send recv init t" shows "\<forall>p. ps (distributed_system.cos channel trans send recv init t (cocomputation.final_i channel trans send recv init t)) p \<noteq> None"
totalI3: fixes f :: "recf" assumes "recfn 3 f"   and "\<And>x y z. eval f [x, y, z] \<down>" shows "Partial_Recursive.total f"
hadd_0_right: fixes x :: "hf" shows "x @+ 0 = x"
mset_msort_key: fixes xs :: "'a list" shows "mset (local.msort_key xs) = mset xs"
l_coset_carrier: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <#\<^bsub>add_monoid G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "y \<in> carrier G"
tCn: shows "t \<in> jkbpC\<^bsub>n\<^esub>"
AssertAllState''_precond': fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<not> \<U> s \<longrightarrow> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<diamond> s" shows "let s' = assert_bound_loop ats s in \<triangle> (\<T> s') \<and> \<nabla> s' \<and> (\<not> \<U> s' \<longrightarrow> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s' \<and> \<diamond> s')"
bst_random_bst: fixes t :: "'a tree"   and A :: "'a set" assumes "t \<in> set_pmf (random_bst A)" shows "bst t"
subst_restrict': fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b"   and env :: "(name, 'b) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "subst t (fmrestrict_fset (frees t) env) = subst t env"
bal_iff_full: fixes t :: "'b tree23" shows "bal t = (\<exists>n. full n t)"
Transaction_Transaction': shows "Transaction = (\<lambda>cm pm views. transaction_merkle.Transaction' ((cm, pm), views))"
insert_select2: fixes l1 :: "'a"   and l2 :: "'a"   and f :: "'a \<Rightarrow> 'b option"   and t :: "'b" assumes "l1 \<noteq> l2" shows "(f(l1 \<mapsto> t)) l2 = f l2"
card_eq_1_singleton: fixes A :: "'a set" assumes "card A = 1" obtains x :: "'a" where "A = {x}"
f_in_funcset: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b" assumes "CLF cl f" shows "f \<in> pset cl \<rightarrow> pset cl"
HSem_merge: fixes ESem :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<rightarrow> 'c"   and \<Gamma> :: "('b \<times> 'a) list"   and \<Delta> :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "has_ignore_fresh_ESem ESem"   and "atom ` domA \<Gamma> \<sharp>* \<Delta>" shows "has_ESem.HSem ESem \<Gamma>\<cdot>(has_ESem.HSem ESem \<Delta>\<cdot>\<rho>) = has_ESem.HSem ESem (\<Gamma> @ \<Delta>)\<cdot>\<rho>"
execute_assert_1: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and h :: "heap" assumes "P x" shows "execute (assert P x) h = Some (x, h)"
intros_2: fixes s :: "s"   and vs :: "v list"   and j :: "nat"   and i :: "nat" shows "\<lparr>s;vs;[$Call j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Callcl (sfunc s i j)]\<rparr>"
Set_project_def: fixes A :: "('b \<times> 'a) set"   and a :: "'b" shows "Set_project A a = {b. (a, b) \<in> A}"
assms: fixes j :: "nat" assumes "j \<le> i" shows "(gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)\<^sup>2 \<le> bnd"
resumption_TERMINAL: fixes a :: "'h" shows "TERMINAL a \<in> resumption"
order_multiplicity_eq: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "order a p = multiplicity [:- a, 1::'b:] p"
equiv_subst_left: fixes cmp :: "'b comparator"   and z :: "'b"   and x :: "'b"   and y :: "'b"   and comp :: "comp" assumes "compare cmp z x = Equiv" shows "(compare cmp z y = comp) = (compare cmp x y = comp)"
an_inj: fixes n :: "nat"   and m :: "nat" assumes "an n = an m" shows "n = m"
rcut2: fixes A :: "fm"   and B :: "fm"   and C :: "fm"   and H :: "fm set"   and D :: "fm" assumes "{A, B} \<turnstile> C"   and "insert C H \<turnstile> D"   and "H \<turnstile> B" shows "insert A H \<turnstile> D"
dbl_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus x = plus x x"
wf_no_loop: fixes R :: "('a \<times> 'a) set" assumes "R O R = {}" shows "wf R"
right_divisibility: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (\<exists>c. a = c * b)"
eq_Abs_Pg: fixes z :: "Pg" obtains g :: "game" where "z = Abs_Pg (eq_game_rel `` {g})"
fst_netgmap_not_none: fixes i :: "nat"   and s :: "'c net_state"   and sr :: "'c \<Rightarrow> 'd \<times> 'e" assumes "i \<in> net_ips s" shows "fst (netgmap sr s) i \<noteq> None"
parIdLeft: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "\<And>Q. (Q \<parallel> \<zero>, Q) \<in> Rel" shows "P \<parallel> \<zero> \<leadsto>[Rel] P"
fls_base_factor_to_fps_zero: shows "fls_base_factor_to_fps 0 = 0"
class_add_is_relevant_entry_rev: fixes C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog"   and i :: "instr"   and pc :: "nat"   and e :: "ex_entry" assumes "is_relevant_entry (class_add P (C, cdec)) i pc e"   and "\<And>cd D'. cd \<in> set P \<Longrightarrow> (fst cd, C) \<notin> (subcls1 P)\<^sup>*"   and "wf_syscls P" shows "is_relevant_entry P i pc e"
conv_dist_comp: 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 bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "conv (times x y) = times (conv y) (conv x)"
assms: shows "(s, s') \<in> R\<^sup>*"
star_mono: fixes L1 :: "'a list set"   and L2 :: "'a list set" assumes "L1 \<subseteq> L2" shows "star L1 \<subseteq> star L2"
simps_1: fixes t :: "'a trm"   and s :: "'a trm" shows "mset_lit (Pos (Eq t s)) = {#t, s#}"
iso_in_homI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>"   and "D.iso f" shows "\<guillemotleft>f : a \<cong>\<^sub>D b\<guillemotright>"
bigthetaD1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "f \<in> O[F](g)"
smult_0_left: fixes p :: "'a poly" shows "Polynomial.smult (0::'a) p = 0"
rtranclp_induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P a"   and "\<And>y z. \<lbrakk>r\<^sup>*\<^sup>* a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
row_mat_of_row: fixes y :: "'a vec" shows "row (mat_of_row y) 0 = y"
lm': shows "Suc (k' + l'') \<le> m'"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>ip = ip r, sn = sn r, rt = rt r, rreqs = rreqs r, store = store r, msg = msg r, data = data r, dests = dests r, pre = pre r, dip = dip r, oip = oip r, hops = hops r, dsn = dsn r, dsk = dsk r, osn = osn r, sip = sip r, \<dots> = state.more r\<rparr>"
simps_4: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "params (And p q) = params p \<union> params q"
rfb_rbd_galois: fixes R :: "('c \<times> 'd) set" shows "bd\<^sub>\<R> R \<stileturn> fb\<^sub>\<R> R"
fder: fixes z :: "complex" assumes "z \<in> S" shows "(f has_field_derivative f' z) (at z)"
cInf_greatest: 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 X :: "'b set"   and z :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq z x" shows "less_eq z (Inf X)"
assms: shows "\<forall>i\<in>fv\<^sub>s\<^sub>e\<^sub>t (set K). i < length T"
NS_refl: fixes s :: "('f, 'b) Term.term" shows "NS s s"
4: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
AECI: shows "AE x in M. 0 < real_cond_exp M Invariants C x"
ladj_def: fixes g :: "'c \<Rightarrow> 'b" shows "ladj g = (\<lambda>x. \<Sqinter> {y. x \<le> g y})"
distinct_20: fixes pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "\<tau>.pi' \<noteq> name1{name2}.pi"
lookupc_unicity: fixes y :: "name"   and a :: "coname"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and x :: "name"   and b :: "coname" assumes "lookupc y a \<theta>_n = Ax x b"   and "x \<sharp> \<theta>_n" shows "y = x"
hom_between_subgroups: fixes A :: "'a set"   and B :: "'a set" assumes "g ` A \<subseteq> B" shows "g \<in> hom (subgroup_generated B A) (subgroup_generated C B)"
intros_5: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<in> i_set_ind"
simps: fixes a :: "'b llist" shows "lfinite a = (a = LNil \<or> (\<exists>xs x. a = LCons x xs \<and> lfinite xs))"
Im1_Hv: fixes v :: "(real, 3) vec" shows "Im1 (Hv v) = v $ 1"
abc_Hoare_haltE: fixes lm1 :: "nat list"   and p :: "abc_inst list"   and lm2 :: "nat list" assumes "{\<lambda>nl. nl = lm1} p {\<lambda>nl. nl = lm2}" shows "\<exists>stp. abc_steps_l (0, lm1) p stp = (length p, lm2)"
igSubstInp_def: fixes MOD :: "('d, 'e, 'f, 'g, 'h, 'i, 'c, 'j, 'k) model_scheme"   and ys :: "'f"   and Y :: "'c"   and y :: "'i"   and inp :: "'b \<Rightarrow> 'c option" shows "igSubstInp MOD ys Y y inp \<equiv> lift (igSubst MOD ys Y y) inp"
lift_spmf_bind_spmf: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'a spmf" shows "lift_spmf (p \<bind> f) = lift_spmf p \<bind> (\<lambda>x. lift_spmf (f x))"
assms_2: shows "\<not> rt1 \<approx>\<^bsub>i\<^esub> rt2"
pred_mono_strong: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a option"   and Pa :: "'a \<Rightarrow> bool" assumes "pred_option P x"   and "\<And>z. \<lbrakk>z \<in> set_option x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_option Pa x"