hcju/isabelleps huggingface dataset:
seq_assign_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and x :: "'a"   and e :: "'c"   and c :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "((((x \<leftarrow> e) ;; c, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "c' = Stop ;; c \<and> mds' = mds \<and> mem' = mem(x := eval\<^sub>A mem e)"
vec_of_list_coeffs_nth': fixes i :: "nat"   and h :: "'d poly"   and u :: "'e poly" assumes "i \<in> {..degree h}"   and "h \<noteq> 0"   and "degree h < degree u" shows "vec_of_list (coeffs h @ replicate (degree u - length (coeffs h)) (0::'d)) $ i = coeff h i"
Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and f :: "'e \<Rightarrow> 'e"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. f s \<in> Q} Basic f Q,A"
delay_monotone: fixes f :: "nat \<Rightarrow> nat option"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and i :: "nat" assumes "f \<in> \<R>"   and "n\<^sub>1 \<le> n\<^sub>2" shows "the (delay i (f \<triangleright> n\<^sub>1)) \<le> the (delay i (f \<triangleright> n\<^sub>2))"
UINF_as_Sup_collect: fixes A :: "'f set"   and f :: "'f \<Rightarrow> ('d, 'e) uexpr" shows "(\<Sqinter> P \<in> A \<bullet> f P) = \<Sqinter>(f ` A)"
cases: fixes x :: "'b atom formula" obtains a :: "predicate" and vs :: "'b list" where "x = Atom (predAtm a vs)"   | va :: "'b" and vb :: "'b" where "x = Atom (Eq va__ vb)"   | "x = \<bottom>"   | v :: "'b atom formula" where "x = \<^bold>\<not> v__"   | v :: "'b atom formula" and va :: "'b atom formula" where "x = vc__ \<^bold>\<and> vaa__"   | v :: "'b atom formula" and va :: "'b atom formula" where "x = vd__ \<^bold>\<or> vab__"   | v :: "'b atom formula" and va :: "'b atom formula" where "x = v \<^bold>\<rightarrow> va"
simps_3: fixes uv :: "typ"   and v :: "variable"   and va :: "typ" shows "eta_reducible (Abs uv (Fv v va)) = eta_reducible (Fv v va)"
smult_closed: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and r :: "'a"   and m :: "'b" assumes "VectorSpace smult V"   and "r \<in> UNIV"   and "m \<in> V" shows "smult r m \<in> V"
insert_remove: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'a measure"   and x :: "'b" assumes "finite A" shows "sup_measure.F g (insert x A) = sup (g x) (sup_measure.F g (A - {x}))"
rep_eq: fixes x :: "'f \<Rightarrow> bool"   and xa :: "('f, 'e) generator" shows "generator (filter_trans x xa) = filter_raw x (generator xa)"
div_mod_iff_dvdm: fixes m :: "int"   and a :: "int poly"   and b :: "int poly" shows "poly_mod.dvdm m a b = (\<exists>q r. b = q * a + smult m r)"
T_insert_def: fixes x :: "'b"   and ts :: "'b tree list" shows "T_insert x ts = T_ins_tree (Node 0 x []) ts + 1"
sub: shows "U' \<subseteq> {w. - pi < Im w \<and> Im w < pi}"
normal_invE_1: fixes N :: "int set" assumes "N \<lhd> Z n" shows "subgroup N (Z n)"
zero_le_imp_eq_int: fixes k :: "Int.int" assumes "0 \<le> k" shows "\<exists>n. k = int n"
complex_cnj_scaleR: fixes r :: "real"   and x :: "complex" shows "cnj (r *\<^sub>R x) = r *\<^sub>R cnj x"
mult_left_one: fixes x :: "'a" shows "(1::'a) * x = x"
ferrack_def: fixes p :: "fm" shows "ferrack p \<equiv> let q = simpfm p; mp = minusinf q; pp = plusinf q in if mp = T \<or> pp = T then T else let U = alluopairs (remdups (uset q)) in decr0 (Parametric_Ferrante_Rackoff.disj mp (Parametric_Ferrante_Rackoff.disj pp (evaldjf (simpfm \<circ> msubst q) U)))"
smap_applicative: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e stream" shows "smap f x = pure f \<diamondop> x"
qmsg_trans: shows "automaton.trans qmsg = seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
known_ptrs_defs: fixes h :: "(_) heap" shows "ShadowRootClass.known_ptrs h = (\<forall>ptr\<in>fset (object_ptr_kinds h). ShadowRootClass.known_ptr ptr)"
subst_clause'_append: fixes A :: "'b ltln list"   and B :: "'b ltln list"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_clause' (A @ B) m = subst_clause' A m \<otimes>\<^sub>l subst_clause' B m"
reduce_simple_not_nil: fixes es :: "e list"   and es' :: "e list" assumes "\<lparr>es\<rparr> \<leadsto> \<lparr>es'\<rparr>" shows "es \<noteq> []"
no_neg_cyc: fixes p :: "'b list" assumes "awalk u p v" shows "\<nexists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0"
dbl_def: fixes x :: "'a" shows "neg_numeral_class.dbl x = x + x"
empty_in_lattice: fixes L :: "'b set set" assumes "lattice L" shows "{} \<in> L"
select_convs_2: fixes dpm :: "(nat \<times> nat \<times> nat) pointermap"   and dcl :: "nat \<times> nat \<times> nat \<Rightarrow> nat option"   and more :: "'a" shows "dcl \<lparr>dpm = dpm, dcl = dcl, \<dots> = more\<rparr> = dcl"
bv_int_bv: fixes i :: "int" shows "bv_to_int (int_to_bv i) = i"
bounded_out: fixes x :: "'v" assumes "x \<in> \<^bold>V - B \<Gamma>" shows "integral\<^sup>N (count_space (\<^bold>E `` {x})) (weight \<Gamma>) < \<top>"
G_validn_def: fixes K :: "nat"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" shows "G_validn K G = (\<forall>C m l A B I. G\<down>(C, m, l) = Some (A, B, I) \<longrightarrow> \<Turnstile>\<^sub>K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I )"
mk_finally_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and x :: "'a ltln" shows "w \<Turnstile>\<^sub>n mk_finally x = w \<Turnstile>\<^sub>n F\<^sub>n x"
mem_Collect_eq: fixes a :: "'ap"   and P :: "'ap \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
sqn_invalidate_in_dests: fixes dests :: "nat \<Rightarrow> nat option"   and ipa :: "nat"   and rsn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dests ipa = Some rsn"   and "ipa \<in> kD rt" shows "sqn (invalidate rt dests) ipa = rsn"
is_ub_empty: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b" assumes "class.po below" shows "po.is_ub below {} u"
special_soundness_def: shows "chaum_ped_sigma.special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub \<longrightarrow> e \<in> challenge_space \<longrightarrow> e' \<in> challenge_space \<longrightarrow> e \<noteq> e' \<longrightarrow> check h a e z \<longrightarrow> check h a e' z' \<longrightarrow> lossless_spmf (ss_adversary h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (ss_adversary h (a, e, z) (a, e', z')). (h, w') \<in> R)"
gb_schema_aux_dgrad_p_set_init: fixes d :: "'a \<Rightarrow> nat"   and sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and m :: "nat"   and rc :: "nat"   and data :: "nat \<times> 'g" assumes "dickson_grading d"   and "struct_spec sel ap ab compl"   and "fst ` (set gs \<union> set bs) \<subseteq> dgrad_p_set d m" shows "fst ` set (gb_schema_aux sel ap ab compl gs (rc, data) (ab gs [] bs data) (ap gs [] [] bs data)) \<subseteq> dgrad_p_set d m"
rel_set_conversep: fixes A :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
F1bd'_Cnotzero: fixes r2 :: "('d \<times> 'd) set" shows "(bd_F1 +c r2, czero) \<notin> ordIso \<and> Card_order (bd_F1 +c r2)"
less_basic_type_def: shows "(<) \<equiv> (\<sqsubset>\<^sub>B)\<^sup>+\<^sup>+"
A: shows "f `# P \<subseteq># M1 + M2"
integral_one_over_log_squared_bound: fixes x :: "real" assumes "4 \<le> x" shows "integral {2..x} (\<lambda>t. 1 / (ln t)\<^sup>2) \<le> sqrt x / (ln 2)\<^sup>2 + 4 * x / (ln x)\<^sup>2"
apply2: fixes i :: "'a" shows "blinfun_apply (f' x0) (blinfun_apply invf i) = i"
rel_option_unfold: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and y :: "'b option" shows "rel_option R x y = (Option.is_none x = Option.is_none y \<and> (\<not> Option.is_none x \<longrightarrow> \<not> Option.is_none y \<longrightarrow> R (the x) (the y)))"
rtranclp_normalize_poly_add_mset: fixes A :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and x :: "char list multiset \<times> int" assumes "normalize_poly_p\<^sup>*\<^sup>* A r" shows "normalize_poly_p\<^sup>*\<^sup>* (add_mset x A) (add_mset x r)"
integrable': fixes x :: "'a" assumes "x \<in> A" shows "set_integrable M {a..} (f x)"
Bossy_CH_BCh: fixes X :: "'a set" shows "Bossy'.CH X = BCh (Xh d1h1) X \<union> BCh (Xh d1h3) X"
Diff_Inter: fixes A :: "'b set"   and S :: "'b set set" shows "A - \<Inter> S = \<Union> {A - s |s. s \<in> S}"
intros_2: fixes S :: "'d \<Rightarrow> bool"   and l :: "'d tree"   and r :: "'d tree"   and v :: "'d" assumes "treesp S l"   and "treesp S r"   and "S v" shows "treesp S \<langle>l, v, r\<rangle>"
tick_count_def: fixes r :: "'b run"   and c :: "clock"   and n :: "nat" shows "tick_count r c n = card {i. i \<le> n \<and> hamlet (Rep_run r i c)}"
r_inv: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
kat_3': fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "am_d x \<cdot> y \<cdot> ad z = a_zero" shows "am_d x \<cdot> y = am_d x \<cdot> y \<cdot> am_d z"
list_relE3: fixes x :: "'d"   and xs :: "'d list"   and l' :: "'e list"   and R :: "('d \<times> 'e) set" assumes "(x # xs, l') \<in> \<langle>R\<rangle>list_rel" obtains x' :: "'e" and xs' :: "'e list" where "l' = x' # xs'" and "(x, x') \<in> R" and "(xs, xs') \<in> \<langle>R\<rangle>list_rel"
dom_reachable: fixes n :: "nat"   and d :: "nat" assumes "reachable n"   and "dominate d n" shows "reachable d"
charpoly_def: fixes A :: "('b, 'c) sq_matrix" shows "charpoly A = Square_Matrix.det (\<^bold>X - \<^bold>C A)"
pWlsInps_2: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsBinp delta binp" shows "gWlsBinp MOD delta (lift (recAbs MOD) binp)"
outstanding_refs_mono_set: fixes sb :: "'b memref list"   and sb' :: "'b memref list"   and P :: "'b memref \<Rightarrow> bool" assumes "set sb \<subseteq> set sb'" shows "outstanding_refs P sb \<subseteq> outstanding_refs P sb'"
us: shows "us = us1 @ berlekamp_factorization_main d divs vs2 n2"
Zfun_zero: fixes F :: "'d filter" shows "Zfun (\<lambda>x. 0::'e) F"
ex_W_cmpserTr3m: fixes m :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "tw_cmpser G m f" shows "\<exists>g. tW_cmpser G (card (f ` {i. i \<le> m}) - 1) g \<and> g ` {i. i \<le> card (f ` {i. i \<le> m}) - 1} = f ` {i. i \<le> m}"
exhaust: fixes y :: "'c \<times> 'd" obtains x1 :: "'c" and x2 :: "'d" where "y = (x1, x2)"
weakly_bisimilar_weak_simulation_step: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and Q :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>"   and "weak_nominal_ts.weakly_bisimilar satisfies transition \<tau> P Q"   and "bn \<alpha> \<sharp>* Q"   and "weak_nominal_ts.weak_transition transition \<tau> P \<alpha> P'" obtains Q' :: "'a" where "weak_nominal_ts.weak_transition transition \<tau> Q \<alpha> Q'" and "weak_nominal_ts.weakly_bisimilar satisfies transition \<tau> P' Q'"
hyps_1: shows "\<langle>c'',s''\<rangle> \<rightarrow> \<langle>c',s'\<rangle>"
F_def: fixes n :: "nat" shows "F n = \<Inter> (E ` {n..})"
mcompose_def: fixes M :: "('e, 'g, 'h) Module_scheme"   and g :: "'i \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'i" shows "mcompose M g f = compose (carrier M) g f"
rep_eq: fixes x :: "\<o>" shows "eval\<o> (\<^bold>\<box>x) = (\<lambda>s w. \<forall>v. eval\<o> x s v)"
intros: fixes x :: "'a"   and r :: "('a \<times> 'a) set" assumes "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r" shows "x \<in> Wellfounded.acc r"
measurable_distr_prob_space: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "(\<lambda>M'. distr M' N f) \<in> prob_algebra M \<rightarrow>\<^sub>M prob_algebra N"
matrix_to_iarray_fst_Gauss_Jordan_upt_k_det_P: fixes k :: "nat"   and A :: "(('d, 'e) vec, 'f) vec" assumes "k < ncols A" shows "fst (Gauss_Jordan_upt_k_det_P A k) = fst (Gauss_Jordan_upt_k_det_P_iarrays (matrix_to_iarray A) k)"
llist_all2_LCons2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and y :: "'b"   and ys :: "'b llist" shows "llist_all2 P xs (LCons y ys) = (\<exists>x xs'. xs = LCons x xs' \<and> P x y \<and> llist_all2 P xs' ys)"
SPLITPRE: shows "finite (set_mset S) \<and> {t} \<subseteq> set_mset S"
sum_Un_ge: fixes M :: "'b set"   and N :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "finite M"   and "finite N"   and "\<forall>B\<in>M \<union> N. 0 < f B" shows "sum f M \<le> sum f (M \<union> N)"
SystemK_PAK: fixes p :: "'a fm" assumes "\<turnstile> p" shows "\<turnstile>\<^sub>! lift p"
ask_get: fixes f :: "'r \<Rightarrow> 's \<Rightarrow> 'm" shows "ask (\<lambda>r. get (f r)) = get (\<lambda>s. ask (\<lambda>r. f r s))"
scale_minus_left: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and x :: "'d" assumes "module scale" shows "scale (- a) x = - scale a x"
Inf_from_q_def: fixes q :: "'q" shows "ground.Inf_from_q q = inference_system.Inf_from (Inf_G_q q)"
slpa_get_proc: fixes cs :: "'edge list"   and as :: "'edge list"   and n :: "'node"   and n' :: "'node" assumes "same_level_path_aux cs as"   and "upd_cs cs as = []"   and "n -as\<rightarrow>* n'"   and "\<forall>c\<in>set cs. valid_edge c" shows "(if cs = [] then get_proc n else get_proc (last (sourcenodes cs))) = get_proc n'"
set_1: fixes x1 :: "const" shows "RID\<^sub>V (CV x1) = {}"
fold_unique: fixes f :: "'d IF1 \<Rightarrow> 'e"   and s1 :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and g :: "'d IF2 \<Rightarrow> 'f"   and s2 :: "('d, 'e, 'f) F2 \<Rightarrow> 'f" assumes "f \<circ> ctor1 = s1 \<circ> F1map id f g"   and "g \<circ> ctor2 = s2 \<circ> F2map id f g" shows "f = fold1 s1 s2 \<and> g = fold2 s1 s2"
reflect_poly_at_0_eq_0_iff: fixes p :: "'a poly" shows "(poly (reflect_poly p) (0::'a) = (0::'a)) = (p = 0)"
compare_is_comparator_of: fixes compare :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.compare_order compare less_eq less" shows "compare = linorder.comparator_of less"
mat2: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and err :: "v error_result"   and pes :: "(pat \<times> exp0) list" assumes "evaluate ck env s e (s', Rerr err)" shows "evaluate ck env s (Mat e pes) (s', Rerr err)"
u_lam: fixes t :: "lam"   and xs :: "name list"   and t' :: "lam"   and x :: "name" assumes "t \<mapsto> xs,t'" shows "Lam [x].t \<mapsto> x # xs,t'"
run_ask_writer: fixes ask :: "('d \<Rightarrow> 'a) \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> ('b, 'c, 'a) writerT" shows "run_writer (ask_writer ask f) = ask (\<lambda>r. run_writer (f r))"
UN_insert: fixes B :: "'c \<Rightarrow> 'b set"   and a :: "'c"   and A :: "'c set" shows "\<Union> (B ` insert a A) = B a \<union> \<Union> (B ` A)"
fmrel_on_fset_alt_def: fixes S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'d \<rightharpoonup>\<^sub>f 'e"   and n :: "'d \<rightharpoonup>\<^sub>f 'f" shows "fmrel_on_fset S P m n = fBall S (\<lambda>x. rel_option P (fmlookup m x) (fmlookup n x))"
simps_2: fixes hs :: "'b tree"   and uu :: "'b" shows "Pairing_Heap_Tree.del_min \<langle>hs, uu, \<langle>\<rangle>\<rangle> = Pairing_Heap_Tree.pass\<^sub>2 (Pairing_Heap_Tree.pass\<^sub>1 hs)"
antisymmetric_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "antisymmetric A less_eq \<equiv> \<forall>x y. less_eq x y \<longrightarrow> less_eq y x \<longrightarrow> x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> x = y"
OP_def: fixes x :: "'b" shows "OP x \<equiv> x"
bell10_def: shows "|\<beta>\<^sub>1\<^sub>0\<rangle> \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m |vec 4 (\<lambda>i. if i = 0 then 1 else if i = 3 then - 1 else 0)\<rangle>"
linear_uminus: shows "Vector_Spaces.linear (*s) (*s) uminus"
wp_loopI: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "('a \<times> 'a) set" assumes "\<lceil>P\<rceil> \<subseteq> \<lceil>I\<rceil>"   and "\<lceil>I\<rceil> \<subseteq> \<lceil>Q\<rceil>"   and "\<lceil>I\<rceil> \<subseteq> wp R \<lceil>I\<rceil>" shows "\<lceil>P\<rceil> \<subseteq> wp (LOOP R INV I ) \<lceil>Q\<rceil>"
d_IN_def: fixes g :: "'c \<times> 'c \<Rightarrow> ennreal"   and y :: "'c" shows "d_IN g y = (\<Sum>\<^sup>+ x. g (x, y))"
generate_topology_Union: fixes I :: "'b set"   and S :: "'c set set"   and K :: "'b \<Rightarrow> 'c set" assumes "\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)" shows "generate_topology S (\<Union> (K ` I))"
wf_on_imp_nf_ex: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" assumes "wfp_on r\<inverse>\<inverse> A"   and "relation.dw_closed r A"   and "a \<in> A" obtains b :: "'b" where "r\<^sup>*\<^sup>* a b" and "relation.is_final r b"
rev_H_injD: fixes x :: "'b"   and z :: "'b"   and y :: "'b" assumes "rev_H' x = z"   and "rev_H' y = z"   and "x \<noteq> y" shows "False"
cancel_separate_5: fixes w :: "'b"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "bounded_distrib_lattice_class.comp_inf.coreflexive (w \<sqinter> id z)" shows "w \<sqinter> (x \<sqinter> (y \<sqinter> z)) \<le> y"
rt_fresh_as_trans: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2"   and "rt2 \<approx>\<^bsub>dip\<^esub> rt3" shows "rt1 \<approx>\<^bsub>dip\<^esub> rt3"
neg0: shows "\<forall>t<0. nth_derivative (Suc k) f t 1 = 0"
set_shadow_root_get_child_nodes_is_l_set_shadow_root_get_child_nodes: shows "l_set_shadow_root_get_child_nodes ShadowRootClass.type_wf set_shadow_root set_shadow_root_locs ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs"
mgu\<^sub>t\<^sub>s_mgu\<^sub>l\<^sub>s: fixes \<theta> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" assumes "mgu\<^sub>t\<^sub>s \<theta> (literal_to_term ` L)" shows "mgu\<^sub>l\<^sub>s \<theta> L"
closed_simple_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "closed (path_image g)"
F_eq: shows "F g S = F (\<lambda>x. h (j x)) S"
WCFG_LAssSkip: fixes V :: "char list"   and e :: "expr" shows "V:=e \<turnstile> (_ 1 _) -\<Up>id\<rightarrow> (_Exit_)"
run_tick_count_suc: fixes r :: "'b run"   and c :: "clock"   and n :: "nat" shows "#\<^sub>\<le> r c Suc n = (if hamlet (Rep_run r (Suc n) c) then Suc #\<^sub>\<le> r c n else #\<^sub>\<le> r c n)"
comm_groupE_5: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "comm_group G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
language_state_elim: fixes w :: "('d \<times> 'e) list"   and M :: "('d, 'e, 'f) FSM"   and q :: "'f" assumes "w \<in> LS M q" obtains r :: "'f list" where "path M (w || r) q" and "length w = length r"
assms: fixes zs :: "'a list"   and ws :: "'b list"   and n :: "nat" assumes "length zs = length ws"   and "n = min (length xs) (length ys)"   and "zs = take n xs"   and "ws = take n ys" shows "P (zip zs ws)"
1: shows "periodic_arithmetic (\<lambda>r. \<chi> r * unity_root n (int (r * k))) n"
sum_nonpos_eq_0_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<forall>x\<in>A. f x \<le> (0::'b)" shows "(sum f A = (0::'b)) = (\<forall>x\<in>A. f x = (0::'b))"
map_cong: fixes x :: "'a blindable\<^sub>h"   and ya :: "'a blindable\<^sub>h"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> set_blindable\<^sub>h ya \<Longrightarrow> f z = g z" shows "map_blindable\<^sub>h f x = map_blindable\<^sub>h g ya"
cauchy_ine\<^sub>N_def: fixes N :: "'b quasinorm"   and u :: "nat \<Rightarrow> 'b" shows "cauchy_ine\<^sub>N N u = (\<forall>e>0. \<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. eNorm N (u n - u m) < e)"
distinct_5: fixes x21 :: "'a tree23"   and x22 :: "int \<times> 'a"   and x23 :: "'a tree23"   and x31 :: "'a tree23"   and x32 :: "int \<times> 'a"   and x33 :: "'a tree23"   and x34 :: "int \<times> 'a"   and x35 :: "'a tree23" shows "Branch2 x21 x22 x23 \<noteq> Branch3 x31 x32 x33 x34 x35"