hcju/isabelleps huggingface dataset:
elims: fixes x :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and y :: "nat" assumes "bheight x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and xa :: "'c \<times> 'd" and c :: "tcolor" and uu_ :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (xa, c, uu_), r\<rangle>" and "y = (if c = Black then bheight l + 1 else bheight l)"
fold_graph_fold: fixes A :: "'a\<^sub>m set"   and z :: "'a\<^sub>m option" assumes "finite A" shows "fold_graph (\<lambda>x yo. yo \<bind> m x) z A (Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z A)"
upred_eq_false: fixes p :: "'a upred" shows "(p =\<^sub>u false) = (\<not> p)"
path_independent_onI2: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "f_range_on A f"   and "Chernoff_on A f"   and "consistency_on A f" shows "path_independent_on A f"
case_2: fixes f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<sigma> \<Rightarrow> 'a"   and x2 :: "\<sigma>" shows "(case \<sigma>\<upsilon> x2 of \<omega>\<upsilon> x \<Rightarrow> f1 x | \<sigma>\<upsilon> x \<Rightarrow> f2 x) = f2 x2"
append_rows_nth_2: fixes A :: "'b mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'b mat"   and nr2 :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc"   and "nr1 \<le> i"   and "i < nr1 + nr2" shows "row (A @\<^sub>r B) i = row B (i - nr1)"
inf_bot_right: fixes x :: "'k multiset" shows "x \<inter># {#} = {#}"
1: shows "x\<^sup>\<star> * z\<^sup>+ * y = z\<^sup>+ * y"
assms: shows "b \<in> arcs H \<inter> G.face_cycle_set a"
cos_2npi: fixes n :: "nat" shows "cos (2 * real n * pi) = 1"
abs_eq: fixes xa :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a ptrm" shows "infer xa (abs_trm x) = ptrm_infer_type xa x"
Entry_path_High_path: fixes as :: "'edge list"   and n :: "'node" assumes "(_Entry_) -as\<rightarrow>* n"   and "inner_node n" obtains a' :: "'edge" and as' :: "'edge list" where "as = a' # as'" and "(_High_) -as'\<rightarrow>* n" and "kind a' = (\<lambda>s. True)\<^sub>\<surd>"
szip_smap_both: fixes f :: "'e \<Rightarrow> 'c"   and xs :: "'e stream"   and g :: "'f \<Rightarrow> 'd"   and ys :: "'f stream" shows "smap f xs ||| smap g ys = smap (map_prod f g) (xs ||| ys)"
emb_step_at_left: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at [] dir.Left (App t s) = t"
filter_eq_iff: fixes F :: "'a filter"   and F' :: "'a filter" shows "(F = F') = (\<forall>P. eventually P F = eventually P F')"
simps_2: fixes w :: "'a list"   and n :: "nat" shows "Lyndon_rec w (Suc n) = (if w <lex rotate (Suc n) w then Lyndon_rec w n else False)"
induct: fixes P :: "nat \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat list" assumes "\<And>n. P n []"   and "\<And>n p ps. \<lbrakk>\<lbrakk>n \<noteq> 0; 1 < p \<and> p dvd n\<rbrakk> \<Longrightarrow> P (n div p) (p # ps); \<lbrakk>n \<noteq> 0; \<not> (1 < p \<and> p dvd n)\<rbrakk> \<Longrightarrow> P n ps\<rbrakk> \<Longrightarrow> P n (p # ps)" shows "P a0 a1"
fls_divide_shift_both_nonzero: fixes f :: "'a fls"   and g :: "'a fls"   and n :: "int"   and m :: "int" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "fls_shift n f / fls_shift m g = fls_shift (n - m) (f / g)"
cut_ge_greater_iprev_conv: fixes t :: "nat"   and I :: "nat set" assumes "t \<in> I"   and "t \<noteq> iMin I" shows "I \<down>\<ge> t = I \<down>> iprev t I"
Suc_pred: fixes n :: "nat" assumes "0 < n" shows "Suc (n - Suc 0) = n"
dd_2: shows "MP_Rel D (of_int_poly (Mp D))"
estep_Id: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and m :: "nat"   and n :: "nat"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((recf.Id m n, xs, ls) # fs, rv)"   and "recfn (length xs) (fst (hd (fst c)))" shows "estep (encode_config c) = encode_config (step c)"
lconf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option" shows "P,h \<turnstile> l (:\<le>) E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
getRel_def: fixes l :: "'e"   and G :: "('e, 'd) labeled_graph" shows "getRel l G = {(x, y). (l, x, y) \<in> edges G}"
mat_col_eqI: fixes B :: "'b mat"   and A :: "'b mat" assumes "\<And>i. i < dim_col B \<Longrightarrow> col A i = col B i"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
psimps_2: fixes uv :: "'a list"   and p :: "(real, 'a) vec" assumes "build_dom (uv, [p])" shows "build uv [p] = Leaf p"
resInputCases: fixes \<Psi> :: "'b"   and x :: "name"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> N"   and "x \<sharp> P'"   and "\<And>P'. \<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P' \<Longrightarrow> Prop (\<lparr>\<nu>x\<rparr>P')" shows "Prop P'"
w_cmpser_ns: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and i :: "nat" assumes "Group G"   and "0 < r"   and "w_cmpser G r f"   and "i \<le> r - 1" shows "\<natural>\<^bsub>G\<^esub>f i \<triangleright> f (Suc i)"
simps: fixes f :: "heap \<Rightarrow> ('a \<times> heap) option" shows "execute (Heap.Heap f) = f"
x_diff_y: shows "xs \<noteq> ys \<or> x \<noteq> y"
I_qelim_anormal: fixes qe :: "'a list \<Rightarrow> 'a fm"   and as :: "'a list"   and xs :: "'b list" assumes "\<And>xs as. \<forall>a\<in>set as. depends\<^sub>0 a \<and> anormal a \<Longrightarrow> is_dnf_qe qe as"   and "\<forall>a\<in>set as. anormal a" shows "I (qelim qe as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>a a (x # xs))"
simps: fixes a :: "'e"   and G :: "('e, 'f) monoid_scheme"   and H :: "'e set" shows "(a \<in> generate G H) = (a = \<one>\<^bsub>G\<^esub> \<or> (\<exists>h. a = h \<and> h \<in> H) \<or> (\<exists>h. a = inv\<^bsub>G\<^esub> h \<and> h \<in> H) \<or> (\<exists>h1 h2. a = h1 \<otimes>\<^bsub>G\<^esub> h2 \<and> h1 \<in> generate G H \<and> h2 \<in> generate G H))"
add: fixes f :: "'h \<Rightarrow> 'i"   and x :: "'h"   and y :: "'h" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
Ord_finite_Sup: fixes A :: "V set" assumes "finite A"   and "A \<subseteq> ON"   and "A \<noteq> {}" shows "\<Squnion> A \<in> A"
linear_spans_image: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec"   and V :: "('c, 'd) vec set"   and B :: "('c, 'd) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "V \<subseteq> vec.span B" shows "f ` V \<subseteq> vec.span (f ` B)"
crit_pair_swap: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "crit_pair p q = (snd (crit_pair q p), fst (crit_pair q p))"
set_shift_vars: fixes vs :: "nat list" shows "set (shift_vars vs) = shift_var_set (set vs)"
edges_are_in_V_2: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Digraph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "w \<in> V\<^bsub>G\<^esub>"
intro: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "comm_monoid f z" shows "comm_monoid_fun f z"
lemAxisIsLine: fixes x :: "'a Point"   and y :: "'a Point"   and z :: "'a Point" assumes "onAxisT x"   and "onAxisT y"   and "onAxisT z"   and "x \<noteq> y"   and "y \<noteq> z"   and "z \<noteq> x" shows "collinear x y z"
remove_heap_is_wellformed_preserved: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "type_wf h"   and "known_ptrs h"   and "heap_is_wellformed h"   and "h \<turnstile> remove child \<rightarrow>\<^sub>h h'" shows "heap_is_wellformed h'"
point_def: shows "point \<equiv> fold (\<lambda>single. subst_term [single]) (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) B"
main: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> set xs_ys" shows "length (filter (\<lambda>x. x \<noteq> a) (map fst xs_ys)) \<le> length xs_ys - 1"
rot_circle_cube_valid_two_cube: shows "valid_two_cube rot_circle_cube"
var_regular_def: fixes uu :: "'a" shows "var_regular uu \<equiv> regular |UNIV|"
one_strongly_connected_iff: fixes x :: "'a" shows "one_strongly_connected x = (connected x \<and> many_strongly_connected x)"
alpha_Tree_valid_Tree: fixes t1 :: "('b, 'pred, 'act) Tree"   and t2 :: "('b, 'pred, 'act) Tree"   and P :: "'state" assumes "t1 =\<^sub>\<alpha> t2" shows "valid_Tree P t1 = valid_Tree P t2"
fps_neg_nth: fixes f :: "'a fps"   and n :: "nat" shows "fps_nth (- f) n = - fps_nth f n"
invar_cons_down: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.invar (t # bq)" shows "BinomialHeapStruc.invar bq"
simps_43: shows "cospan_shape.comp cospan_shape.AA cospan_shape.BT = cospan_shape.Null"
rel_inject_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x2 :: "'b"   and y2 :: "'c" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
WTrtSCall: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts' :: "ty list"   and sfs :: "sfields"   and vs :: "val list" assumes "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "P,E,h,sh \<turnstile> es [:] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts"   and "M = clinit \<longrightarrow> sh D = \<lfloor>(sfs, Processing)\<rfloor> \<and> es = map Val vs" shows "P,E,h,sh \<turnstile> C\<bullet>\<^sub>sM(es) : T"
if_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 b :: "'d"   and p :: "('a, 'c, 'd) Stmt"   and q :: "('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 eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((Stmt.If b p q, mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B" obtains "c' = p" and "mem' = mem" and "mds' = mds" and "eval\<^sub>B mem b"   | "c' = q" and "mem' = mem" and "mds' = mds" and "\<not> eval\<^sub>B mem b"
lub_take: shows "(\<Squnion>n. ValK_take n) = ID"
assms_2: shows "p \<noteq> 0\<^sub>p"
closure_subs: fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<subseteq> V" shows "Z \<subseteq> Closure\<^sub>\<alpha> Z"
dom_simp: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.dom f = local.dom f"
is_renaming_def: fixes id_subst :: "'b"   and comp_subst :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<sigma> :: "'b" shows "substitution_ops.is_renaming id_subst comp_subst \<sigma> = (\<exists>\<tau>. comp_subst \<sigma> \<tau> = id_subst)"
leq_zero_imp: fixes x :: "nat + 'a Bonk_Schramm_extension_unfolded set" assumes "(x, zero wo_aux) \<in> wo_aux" shows "x = zero wo_aux"
gen_dfs_post_imp_complete: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and V0 :: "'a set"   and u0 :: "'a"   and V :: "'a set" assumes "gen_dfs_pre E U {} V0 u0"   and "gen_dfs_post E U {} V0 u0 V False" shows "V0 \<union> E\<^sup>* `` {u0} \<subseteq> V"
ovalid_def: fixes P :: "'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f option"   and Q :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "ovalid P f Q \<equiv> \<forall>s r. P s \<and> f s = Some r \<longrightarrow> Q r s"
pref_sing_power: fixes w :: "'b list"   and a :: "'b"   and m :: "nat" assumes "w \<le>p [a] \<^sup>@ m" shows "w = [a] \<^sup>@ \<^bold>|w\<^bold>|"
valid_characteristic_formula: fixes P' :: "'state" shows "P' \<Turnstile> characteristic_formula P = (\<forall>x\<in>{distinguishing_formula P Q |Q. \<not> P =\<cdot> Q}. P' \<Turnstile> x)"
inf_idem: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) x = x"
foldD_insert: fixes B :: "'e set"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e"   and e :: "'f" assumes "LCD B D f"   and "finite A"   and "x \<notin> A"   and "x \<in> B"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D f e (insert x A) = f x (foldD D f e A)"
try_bind_spmf_lossless2': fixes g :: "'a \<Rightarrow> 'b spmf"   and h :: "'a \<Rightarrow> 'b spmf"   and f :: "'a \<Rightarrow> 'b spmf"   and q :: "'b spmf"   and p :: "'a spmf" assumes "NO_MATCH (\<lambda>x. TRY g x ELSE h x) f"   and "lossless_spmf q" shows "TRY p \<bind> f ELSE q = TRY p \<bind> (\<lambda>x. TRY f x ELSE q) ELSE q"
newconf_lemma: fixes m :: "nat"   and c :: "nat" shows "rec_exec rec_newconf [m, c] = newconf m c"
compact_sequence_with_limit: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "f \<longlonglongrightarrow> l" shows "compact (insert l (range f))"
coreflexive_transitive: fixes x :: "'a" assumes "coreflexive x" shows "transitive x"
analz_insert_freshK: fixes evs :: "event list"   and KAB :: "nat"   and K :: "nat" assumes "evs \<in> recur"   and "KAB \<notin> range shrK" shows "(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = (K = KAB \<or> Key K \<in> analz (knows Spy evs))"
sigma_sets_eq: fixes \<Omega> :: "'a set"   and M :: "'a set set" assumes "sigma_algebra \<Omega> M" shows "sigma_sets \<Omega> M = M"
Cons_1: fixes ys :: "'b list" assumes "length xs = length ys" shows "zip xs ys = map (\<lambda>i. (xs ! i, ys ! i)) [0..<length ys]"
ordIso_iff_ordLeq: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" shows "ordIso2 r r' = (ordLeq3 r r' \<and> ordLeq3 r' r)"
down_cast_same: assumes "is_down UCAST('a \<rightarrow> 'b)" shows "UCAST('a \<rightarrow> 'b) = SCAST('a \<rightarrow> 'b)"
assms: shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi"
locally_full: fixes f :: "'a"   and f' :: "'a"   and \<nu> :: "('a S.E.term, 'a) S.arr" assumes "ide f"   and "ide f'"   and "src f = src f'"   and "trg f = trg f'"   and "\<guillemotleft>\<nu> : S.UP f \<Rightarrow>\<^sub>S S.UP f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow> f'\<guillemotright> \<and> S.UP \<mu> = \<nu>"
lookup_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (monom_mult c t p) v = (if t adds\<^sub>p v then c * lookup p (v \<ominus> t) else (0::'c))"
maxsimp_connect: fixes x :: "'a set"   and y :: "'a set" assumes "x \<noteq> y"   and "chamber x"   and "chamber y" shows "\<exists>xs. gallery (x # xs @ [y])"
elementary_category_with_binary_products_axioms: shows "elementary_category_with_binary_products (\<cdot>) pr0 pr1"
reverse_reverse: fixes xs :: "['b]" assumes "finite_list xs" shows "reverse\<cdot>(reverse\<cdot>xs) = xs"
prems_9: shows "\<alpha> = Cantor_sum \<alpha>s ms"
log_inverse_eq: fixes B :: "real" shows "log b (inverse B) = (if 0 < B then - log b B else log b 0)"
ra_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y ; 1) ; z = x ; z \<cdot> y ; 1"
accmodi_methdMembr_simp: fixes m :: "qtname \<times> mdecl" shows "accmodi (methdMembr m) = accmodi m"
pos_le_divideR_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 scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "real"   and a :: "'c"   and b :: "'c" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less_eq a (scaleR (inverse c) b) = less_eq (scaleR c a) b"
evContainsEx_repeat: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and body :: "formula" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f (i, FAll Neg body))" shows "EV (contains f (Suc i, FAll Neg body))"
roofed_circ_def: fixes \<Gamma> :: "('c, 'd) web_scheme"   and S :: "'c set" shows "RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S = RF\<^bsub>\<Gamma>\<^esub> S - \<E>\<^bsub>\<Gamma>\<^esub> S"
funpow_0: fixes f :: "'g \<Rightarrow> 'g"   and x :: "'g" shows "(f ^^ 0) x = x"
integral_completion: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "integral\<^sup>L (completion M) f = integral\<^sup>L M f"
assms_2: shows "b \<in>\<^sub>\<circ> Vset \<alpha>"
simps_3: fixes i :: "nat"   and l :: "nat bdd"   and r :: "nat bdd"   and T :: "bool list list" shows "check_eq (Leaf i) (Branch l r) T = (check_eq (Leaf i) l T \<and> check_eq (Leaf i) r T)"
unop_bij_def: fixes A :: "V"   and f :: "V" shows "unop_bij A f \<equiv> nop_bij A (1\<^sub>\<nat>) f"
returns_result_select_result: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog" assumes "h \<turnstile> ok f" shows "h \<turnstile> f \<rightarrow>\<^sub>r |h \<turnstile> f|\<^sub>r"
finite_surj_inj: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "finite A"   and "A \<subseteq> f ` A" shows "inj_on f A"
butterfly_eq_proj: fixes x :: "'a" assumes "norm x = 1" shows "selfbutter x = proj x"
nv_3: fixes w :: "'a vec" assumes "w \<in> D" shows "w \<bullet> normal_vector D = (0::'a)"
inverse_arrow_unique: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "C.inverse_arrows f g"   and "C.inverse_arrows f g'" shows "g = g'"
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, rreqid = rreqid 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>"
evalC_sum_list: fixes b :: "nat"   and ns :: "C list" shows "evalC b (C ns) = (\<Sum>n\<leftarrow>ns. b ^ evalC b n)"
results_in_mono_2: fixes s1 :: "'sum"   and s2 :: "'sum"   and t :: "'t" assumes "s1 \<le> s2" shows "results_in t s1 \<le> results_in t s2"
intro: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>x y. (dist x y = 0) = (x = y)"   and "\<And>x z y. dist x z \<le> dist y x + dist y z" shows "metric dist"
arr_dom_iff_arr: fixes f :: "'c \<times> 'c" shows "C.VV.arr (C.VV.dom f) = C.VV.arr f"
compl_coset: fixes xs :: "'c list" shows "- List.coset xs = set xs"
tree_induct: fixes phi :: "('a, 'b) tree \<Rightarrow> bool"   and T :: "('a, 'b) tree" assumes "\<And>inp binp. \<lbrakk>liftAll phi inp; liftAll phi binp\<rbrakk> \<Longrightarrow> phi (Branch inp binp)" shows "phi T"
nsum_minus: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier M" shows "-\<^sub>a \<Sigma>\<^sub>e M f n = \<Sigma>\<^sub>e M (\<lambda>x\<in>{j. j \<le> n}. -\<^sub>a f x) n"
null_sets_cbox_Diff_box: fixes a :: "'c"   and b :: "'c" shows "cbox a b - box a b \<in> null_sets lborel"
f0x: shows "\<forall>x. f (0::'a) x = x"
polyneg_normh: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p n" shows "isnpolyh (~\<^sub>p p) n"
assms: shows "\<Gamma> \<turnstile> Unit : \<tau>"
w_set': shows "\<forall>x\<in>set w. x \<in> {0..<n}"
alts'_subset: shows "{a, b, c} \<subseteq> alts"
tl_def: fixes list :: "'e list" shows "tl list = (case list of [] \<Rightarrow> [] | x21 # x22 \<Rightarrow> x22)"
large_m_def: fixes u :: "int poly"   and vs :: "'b poly list" shows "large_m u vs = (\<forall>v n. v dvd u \<longrightarrow> degree v \<le> degree_bound vs \<longrightarrow> 2 * \<bar>coeff v n\<bar> < m)"
edge_reappears_at_increased_labeling: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and p :: "pr_operation' list"   and f' :: "nat \<times> nat \<Rightarrow> 'capacity"   and l' :: "nat \<Rightarrow> nat"   and v :: "nat"   and u :: "nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts'"   and "l v + 1 \<le> l u"   and "(u, v) \<notin> cfE_of f"   and "(u, v) \<in> cfE_of f'" shows "l v < l' v"
Spec: fixes s :: "'d"   and t :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "(s, t) \<in> r" shows "\<Gamma>\<turnstile> (Spec r # cs, css, Normal s) \<rightarrow> (cs, css, Normal t)"
irreducible_degree_field: fixes p :: "'a poly" assumes "irreducible p" shows "0 < degree p"
Seq_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and c1 :: "com"   and c2 :: "com"   and s1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and s3 :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (c1;; c2, s1) \<Rightarrow> s3 = (\<exists>s\<^sub>2. \<pi>: (c1, s1) \<Rightarrow> s\<^sub>2 \<and> \<pi>: (c2, s\<^sub>2) \<Rightarrow> s3)"
sup_left_dist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<squnion> - y \<sqinter> - z = (- x \<squnion> - y) \<sqinter> (- x \<squnion> - z)"
simps: fixes s :: "expr set" shows "order_radicals s = {y. \<exists>x\<in>s. y = order x}"
3: assumes "b \<noteq> c"   and "a = b" shows "\<exists>f. f b * b + f c * c = x * b + y * b + z * c"
shadows_consts_def: fixes C_info :: "(name, dt_def) fmap"   and heads :: "name fset"   and t :: "'a" shows "pre_constants.shadows_consts C_info heads t = (\<not> fdisjnt (pre_constants.all_consts C_info heads) (all_frees t))"
0: shows "to_fun (shift 2 (T\<^bsub>b\<^esub> g)) (a \<ominus> b) = to_fun (shift 2 (T\<^bsub>b\<^esub> g)) (a \<ominus> b)"
head_if: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {m..n} = (if n < m then 0::'b else sum g {m..<n} + g n)"
genby_uminus_closed: fixes g :: "'b"   and S :: "'b set" assumes "g \<in> \<langle>S\<rangle>" shows "- g \<in> \<langle>S\<rangle>"
cases: fixes x :: "val \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set)" obtains uu_ :: "nat" and i :: "int" and \<beta> :: "nat \<Rightarrow> 'a option" and ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" where "x = (C uu_ i, \<beta>__, ve__)"   | prim :: "prim" and \<beta> :: "nat \<Rightarrow> 'a option" and ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" where "x = (P prim, \<beta>'__, vea__)"   | uv_ :: "nat" and var :: "nat \<times> char list" and \<beta> :: "nat \<Rightarrow> 'a option" and ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" where "x = (R uv_ var, \<beta>''__, veb__)"   | lam :: "lambda" and \<beta> :: "nat \<Rightarrow> 'a option" and ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" where "x = (L lam, \<beta>, ve)"
complete_insert: fixes t :: "'b tree23"   and a :: "'b" assumes "complete t" shows "complete (insert a t)"
distinct_201: fixes x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term"   and x11 :: "Syntax.term" shows "Syntax.Let x81 x82 x83 \<noteq> Prj1 x11"