hcju/isabelleps huggingface dataset:
\<tau>diverge_simulation1: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and \<mu>1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and \<mu>2 :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i" assumes "delay_bisimulation_measure trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 \<mu>1 \<mu>2"   and "\<tau>trsys.\<tau>diverge trsys1 \<tau>move1 s1"   and "bisim s1 s2" shows "\<tau>trsys.\<tau>diverge trsys2 \<tau>move2 s2"
assms_1: shows "N \<subseteq>\<^sub>N N1"
lunit'_simps_1: fixes f :: "'a" assumes "S.ide f" shows "S.arr (S.lunit' f)"
distinct_9: fixes x2 :: "'b match_expr" shows "MatchNot x2 \<noteq> MatchAny"
mk_eqcl_fst_Some: fixes i :: "nat"   and xs :: "nat option list"   and k :: "nat"   and zs :: "nat list"   and ii :: "nat"   and T :: "bool list list" assumes "i < length xs"   and "k < length zs" shows "(fst (mk_eqcl xs zs ii T) ! i = k) = (xs ! i = Some k)"
prems_2: shows "b # v \<preceq>\<^sub>F w"
hom_power: fixes x :: "rat"   and n :: "nat" shows "of_rat (x ^ n) = of_rat x ^ n"
E_Ctxt: fixes t :: "trm"   and t' :: "trm"   and E :: "trm \<Rightarrow> trm" assumes "t \<longmapsto>\<^sub>c t'"   and "E \<in> ctxt" shows "E t \<longmapsto>\<^sub>c E t'"
assms: shows "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
WTrts_hext_mono: fixes h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and es :: "'addr expr list"   and Ts :: "ty list"   and h' :: "'heap" assumes "P,E,h \<turnstile> es [:] Ts"   and "h \<unlhd> h'" shows "P,E,h' \<turnstile> es [:] Ts"
Times_mset_image_mset1: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c multiset"   and B :: "'b multiset" shows "image_mset f A \<times># B = {#(f a, b). (a, b) \<in># A \<times># B#}"
i_Exec_Comp_Stream_Acc_LocalState_Init_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_Init k localState trans_fun input c \<equiv> [localState c] \<frown> i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c"
univalent_finite_1: fixes R :: "('a \<times> 'b) set" assumes "univalent R" shows "card (Domain R) = card R"
Inf_lower: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "\<Sqinter>A \<le> x"
core_nhds_iff: fixes U :: "'b set"   and T :: "'b set set"   and x :: "'b" assumes "U \<subseteq> carrier\<^bsub>T\<^esub>" shows "(x \<in> core\<^bsub>T\<^esub> U) = (U \<in> nhds\<^bsub>T\<^esub> x)"
sublist_app: fixes i1 :: "nat"   and i2 :: "nat"   and i3 :: "nat"   and xs :: "'c list" assumes "i1 \<le> i2"   and "i2 \<le> i3" shows "WB_Sort.sublist xs i1 i2 @ WB_Sort.sublist xs (Suc i2) i3 = WB_Sort.sublist xs i1 i3"
count_def: fixes cs :: "('b \<Rightarrow> bool) list"   and a :: "'b"   and k :: "nat" shows "count cs a k \<equiv> if k < length cs then if (cs ! k) a then Suc k mod length cs else k else if cs = [] then k else 0"
fully_faithful_and_essentially_surjective_functor_axioms: shows "fully_faithful_and_essentially_surjective_functor hhom\<^sub>C.comp hhom\<^sub>D.comp F\<^sub>1"
poly_higher_pderiv_aux3': fixes n :: "nat"   and k :: "nat"   and c :: "'a"   and p :: "'a poly" assumes "n \<le> k"   and "[:- c, 1::'a:] ^ n dvd p" shows "fact n dvd poly ((pderiv ^^ k) p) c"
assms: shows "desargues_config A B C A' B' C' M N P R"
simps_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(pnat x1 = pnat y1) = (x1 = y1)"
mod: shows "mod_hom K V1 (direct_sum V1 V2) (inj1 V1 V2)"
fresh2_notIn: fixes xs :: "nat set"   and x :: "nat"   and y :: "nat" assumes "finite xs" shows "fresh2 xs x y \<notin> xs"
one_in_totatives: fixes n :: "nat" assumes "0 < n" shows "Suc 0 \<in> totatives n"
group_by_single: fixes a :: "'a"   and x :: "'b set"   and f :: "'c \<Rightarrow> 'a \<times> 'b"   and s :: "'c set"   and y :: "'b set" assumes "(a, x) \<in> group_by f s"   and "(a, y) \<in> group_by f s" shows "x = y"
matrix_sum_mult_right: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and d :: "nat"   and A :: "'a mat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "A \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>k. f k * A) n = matrix_sum d f n * A"
X: shows "X \<in> M \<rightarrow>\<^sub>M N"
partitions_remove1_bounds: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat"   and i :: "nat" assumes "p partitions n"   and "0 < p k"   and "(p(k := p k - 1)) i \<noteq> 0" shows "1 \<le> i \<and> i \<le> n - k"
SUP_upper2: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and i :: "'e"   and A :: "'e set"   and u :: "'d"   and f :: "'e \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq u (f i)" shows "less_eq u (Sup (f ` A))"
ConsThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and es :: "nat exp list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>throw e' # es,s\<^sub>1\<rangle>"
Mp_const_poly: fixes m :: "int"   and a :: "int" shows "poly_mod.Mp m [:a:] = [:a mod m:]"
interleaves_swap: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "xs \<simeq> {ys, zs, P} = xs \<simeq> {zs, ys, \<lambda>w ws. \<not> P w ws}"
list2listNMT: fixes x :: "'a list"   and sem :: "'a \<Rightarrow> 'b" assumes "x \<noteq> []" shows "map sem x \<noteq> []"
simps_6: fixes \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "eval (Neg \<phi>) \<Gamma> = (\<not> eval \<phi> \<Gamma>)"
l_inv_ex: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<exists>y\<in>carrier F. y \<oplus>\<^bsub>F\<^esub> x = \<zero>\<^bsub>F\<^esub>"
\<Z>_\<omega>_\<alpha>\<omega>: shows "\<omega> \<in>\<^sub>\<circ> \<alpha> + \<omega>"
mds\<^sub>s_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and x :: "Mode" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.mds\<^sub>s x = {}"
ring_hom_closed: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and x :: "'a" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R" shows "h x \<in> carrier S"
abs_eq: fixes x :: "16 word" shows "uint_of_uint16 (Abs_uint16 x) = Abs_uint (ucast x)"
q1: shows "interaction_bounded_by' (\<lambda>_. True) \<A>1 q1"
cmp_simps_4: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "C.dom (\<Phi>\<^sub>H (f, g)) = H f \<star>\<^sub>C H g"
le_MkI_MkI: fixes x :: "int"   and y :: "int" shows "le\<cdot>(MkI\<cdot>x)\<cdot>(MkI\<cdot>y) = (if x \<le> y then TT else FF)"
B: fixes x :: "real" assumes "x \<in> {0..1}" shows "onorm (f' (x0 + x *\<^sub>R a)) \<le> B"
join: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" assumes "rh a = rh b"   and "a \<in> A" shows "\<exists>ab. m a b = Some ab \<and> bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u)"
ticks_imp_ticks_subk: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and k :: "nat"   and c :: "clock" assumes "dilating f sub r"   and "hamlet (Rep_run r k c)" shows "\<exists>k\<^sub>0. f k\<^sub>0 = k \<and> hamlet (Rep_run sub k\<^sub>0 c)"
supp_list_cons: fixes x :: "'b"   and xs :: "'b list" shows "supp (x # xs) = supp x \<union> supp xs"
simps: fixes todo :: "'b"   and fin :: "'a set" shows "mk_rtrancl_main todo fin = (if set_of todo = {} then fin else let (a, tod) = sel todo in if \<exists>b\<in>fin. subsumes b a then mk_rtrancl_main tod fin else mk_rtrancl_main (un (r a) tod) (insert a fin))"
is_cross_ratio_01inf: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and f :: "complex_homo \<Rightarrow> complex_homo" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3"   and "is_moebius f"   and "f z1 = 0\<^sub>h"   and "f z2 = 1\<^sub>h"   and "f z3 = \<infinity>\<^sub>h" shows "f = (\<lambda>z. cross_ratio z z1 z2 z3)"
Seq2_ir_valid: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and d :: "com"   and c' :: "com"   and R :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P c SKIP Q"   and "ir_valid Q d c' R" shows "ir_valid P (c;; d) c' R"
SUCCEED_rule: fixes \<Phi> :: "'c \<Rightarrow> bool" shows "SUCCEED \<le> SPEC \<Phi>"
mult_left_sub_dist_sup_right: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<odot> z \<preceq> x \<odot> (y \<oplus> z)"
coplanar_perm_9: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet B C D A"
diagonal_to_Smith_row_i_preserves_previous_diagonal: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and b :: "'b" assumes "is_bezout_ext bezout"   and "i < min (nrows A) (ncols A)"   and "to_nat a \<notin> set [i + 1..<min (nrows A) (ncols A)]"   and "to_nat a = to_nat b"   and "to_nat a \<noteq> i" shows "Diagonal_to_Smith_row_i A i bezout $ a $ b = A $ a $ b"
diff_diff_cancel: fixes i :: "nat"   and n :: "nat" assumes "i \<le> n" shows "n - (n - i) = i"
degen_is_run_sound: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" assumes "graph_defs.is_run (degeneralize_ext m) r" shows "is_run (fst \<circ> r)"
complex_mult_cnj: fixes z :: "complex" shows "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
V_def: shows "V = {u n / real n |n. N0 < n}"
minimal_imp_isomorphic_to_canonical: fixes M :: "'b dfa" assumes "dfa M"   and "dfa.minimal M" shows "dfa_isomorphism M (MyhillNerode_dfa.DFA (eq_app_right (dfa.language M)) (dfa.language M) (dfa.index_f M)) (dfa.iso M)"
same_append_eq: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" shows "(xs @ ys = xs @ zs) = (ys = zs)"
irreducible\<^sub>d_mD_1: fixes f :: "int poly" assumes "irreducible\<^sub>d_m f" shows "0 < degree_m f"
subst_rename: fixes y :: "name"   and t :: "lam"   and x :: "name"   and s :: "lam" assumes "y \<sharp> t" shows "t[x::=s] = ([(y, x)] \<bullet> t)[y::=s]"
Abinds_append_disjoint: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "domA \<Delta> \<inter> domA \<Gamma> = {}" shows "ABinds (\<Delta> @ \<Gamma>)\<cdot>ae = ABinds \<Delta>\<cdot>ae \<squnion> ABinds \<Gamma>\<cdot>ae"
hyps_2: fixes s :: "char list \<Rightarrow> int" shows "P s \<le> P' s"
D\<^sub>i\<^sub>i: fixes x :: "i" shows "((\<^bold>\<forall>xa. (dom x) \<cdot> xa \<cong> xa \<^bold>\<leftarrow> E ((dom x) \<cdot> xa)) \<^bold>\<and> (\<^bold>\<forall>xa. xa \<cdot> (dom x) \<cong> xa \<^bold>\<leftarrow> E (xa \<cdot> (dom x)))) \<^bold>\<and> x \<cdot> (dom x) \<cong> x \<^bold>\<leftarrow> E x"
2: shows "(- (x ^ 3) + 6 * x\<^sup>2 - 24 * x + 48)\<^sup>2 / 2304 \<le> inverse (exp x)"
necess_cond_2_lhs: shows "int (size {#x \<in># mset_set \<V> \<times># \<B>. fst x \<in> snd x#}) = \<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1))"
has_contour_integral_div: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "(f has_contour_integral i) g" shows "((\<lambda>x. f x / c) has_contour_integral i / c) g"
gprod_2_2_def: fixes R :: "('a \<times> 'a) set" shows "gprod_2_2 R \<equiv> {((a, b), c, d). a = c \<and> (b, d) \<in> R \<or> b = d \<and> (a, c) \<in> R}"
assms: fixes x :: "'b" assumes "x \<in> X0" shows "\<not> g x"
is_dghmD_9: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv (\<FF>\<lparr>ArrMap\<rparr>)"
elims_1: fixes x :: "'b ugraph"   and xa :: "'b"   and xb :: "('b \<times> 'b) list"   and xc :: "'b"   and y :: "bool" assumes "path x xa xb xc = y" obtains g :: "'b ugraph" and u :: "'b" and v :: "'b" where "x = ga__" and "xa = u__" and "xb = []" and "xc = v" and "y = (u__ = v)"   | g :: "'b ugraph" and u :: "'b" and e :: "'b \<times> 'b" and ps :: "('b \<times> 'b) list" and w :: "'b" where "x = g" and "xa = u" and "xb = e # ps" and "xc = w" and "y = (\<exists>v. e = (u, v) \<and> e \<in> edges g \<and> path g v ps w)"
degree_m_mult_le: fixes f :: "int poly"   and g :: "int poly" shows "degree_m (f * g) \<le> degree_m f + degree_m g"
comp_ide_self: fixes a :: "'a" assumes "C.ide a" shows "a \<cdot>\<^sub>C a = a"
last_in_set: fixes l :: "'a list" assumes "l \<noteq> []" shows "last l \<in> set l"
aux_lemma: fixes x :: "'a"   and xsa :: "'a list" assumes "x \<in> carrier R"   and "set xsa \<subseteq> carrier R" shows "x \<in> finite_extension K (x # xsa)"
par_beta_cases_1: fixes n :: "nat"   and t :: "dB" assumes "Var n => t" obtains "t = Var n"
ord_resolve_rename_ground_inst_sound: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset"   and \<rho>s :: "'s list"   and \<rho> :: "'s"   and I :: "'a set"   and \<eta> :: "'s" assumes "ord_resolve_rename S CAs DA AAs As \<sigma> E"   and "\<rho>s = tl (renamings_apart (DA # CAs))"   and "\<rho> = hd (renamings_apart (DA # CAs))"   and "I \<Turnstile>m mset (CAs \<cdot>\<cdot>cl \<rho>s) \<cdot>cm \<sigma> \<cdot>cm \<eta>"   and "I \<Turnstile> DA \<cdot> \<rho> \<cdot> \<sigma> \<cdot> \<eta>"   and "is_ground_subst \<eta>" shows "I \<Turnstile> E \<cdot> \<eta>"
hyp_echelon: shows "echelon_form_upt_k (echelon_form_of_upt_k A k bezout) (Suc k)"
starset_mem: fixes x :: "'b"   and A :: "'b set" shows "(star_of x \<in> *s* A) = (x \<in> A)"
residual_l_pnat_def: fixes y :: "pnat"   and x :: "pnat" shows "y \<leftarrow> x \<equiv> y - x"
length_suffixes: fixes xs :: "'e list" shows "length (suffixes xs) = Suc (length xs)"
padd_correct: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a \<times> 'a \<times> 'a"   and p\<^sub>2 :: "'a \<times> 'a \<times> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curvep a b p\<^sub>1"   and "on_curvep a b p\<^sub>2" shows "make_affine (padd a p\<^sub>1 p\<^sub>2) = local.add a (make_affine p\<^sub>1) (make_affine p\<^sub>2)"
ext_surjective: fixes x :: "'a point2d_scheme" shows "x = \<lparr>xc = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst point2d_ext_Tuple_Iso \<circ> id)) x, yc = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst point2d_ext_Tuple_Iso \<circ> id)) x, \<dots> = (Record.iso_tuple_snd point2d_ext_Tuple_Iso \<circ> id) x\<rparr>"
correctCompositionDiffLevelsS11opt: shows "correctCompositionDiffLevels sS11opt"
par_ctranE: fixes c :: "'b com option list \<times> 'b"   and c' :: "'b com option list \<times> 'b" assumes "c -pc\<rightarrow> c'" obtains i :: "nat" and Ps :: "'b com option list" and s :: "'b" and r :: "'b com option" and t :: "'b" where "c = (Ps, s)" and "c' = (Ps[i := r], t)" and "i < length Ps" and "(Ps ! i, s) -c\<rightarrow> (r, t)"
exchange: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "a * b ; (c * d) \<le> b ; c * (a ; d)"
mod_ring_eq_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> (=)) (=) (=)"
lookup_default': fixes x :: "'c"   and d :: "'b"   and m :: "('c, 'b) mapping"   and y :: "'c" shows "Mapping.lookup (Mapping.default x d m) y = (if x = y then Some (Mapping.lookup_default d m x) else Mapping.lookup m y)"
rbt_update: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c"   and y :: "'d" assumes "rbt t" shows "rbt (update x y t)"
pickT_pred_unique: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and II :: "nat set"   and n1 :: "nat"   and n2 :: "nat" assumes "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "pickT_pred cl P II n1"   and "pickT_pred cl P II n2" shows "n1 = n2"
eq: shows "(- (1::'a)) ^ Suc h = (\<Prod>i = 0..h. - (1::'a))"
opS: shows "openin (top_of_set (S \<union> T)) S"
perm_simps_4: fixes p :: "perm"   and fm :: "fm" shows "p \<bullet> Neg fm = Neg (p \<bullet> fm)"
fstE: fixes P :: "'b ptrm"   and Y :: "'b ptrm" assumes "PFst P \<approx> Y" obtains B :: "'b ptrm" where "Y = PFst B" and "P \<approx> B"
MakeCatObj: fixes C :: "('a, 'b, 'c) Category_scheme" shows "obj\<^bsub>MakeCat C\<^esub> = obj\<^bsub>C\<^esub>"
nsg_sg: fixes H :: "'a set" assumes "G \<triangleright> H" shows "G \<guillemotright> H "
eval_Mn_convergI: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list"   and z :: "nat" assumes "recfn (length xs) (Mn n f)"   and "eval f (z # xs) \<down>= 0"   and "\<And>y. y < z \<Longrightarrow> eval f (y # xs) \<down>\<noteq> 0" shows "eval (Mn n f) xs \<down>= z"
intensional_simps_9: fixes y :: "'a \<Rightarrow> bool" shows "y = #True \<equiv> y"
substPsi_axioms: shows "substPsi substTerm substAssert substCond"
mkIde_as_mkArr: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set" assumes "A \<subseteq> S.Univ" shows "S.mkArr A A (\<lambda>x. x) = S.mkIde A"
greatest_Above: fixes r :: "('a \<times> 'a) set"   and X :: "'a set" shows "greatest r X = Above r X \<inter> X"
intros: fixes core :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and I :: "'a \<Rightarrow> bool"   and \<I>_adv :: "('c, 'e) \<I>"   and \<I>_usr :: "('d, 'f) \<I>" assumes "\<And>s e s'. \<lbrakk>s' \<in> set_spmf (cpoke core s e); I s\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (cfunc_adv core s x); x \<in> outs_\<I> \<I>_adv; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_adv x \<and> I s'"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (cfunc_usr core s x); x \<in> outs_\<I> \<I>_usr; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_usr x \<and> I s'" shows "WT_core \<I>_adv \<I>_usr I core"
l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes create_document :: "((_) heap, exception, (_) document_ptr) prog" shows "l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document \<equiv> create_document = create_document"
adjoint_pair_antipar_1: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "adjoint_pair f g" shows "ide f"
l_2: shows "(u \<circ> a) \<longlonglongrightarrow> limsup (u \<circ> r)"
mono_post: fixes C :: "'a acom"   and C' :: "'a acom" assumes "C \<le> C'" shows "post C \<le> post C'"
restrict_index_in: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and i :: "nat"   and X :: "'c option list"   and I :: "nat set" assumes "getIJ getIJ"   and "i < length X"   and "i \<in> I" shows "restrict I X ! i = X ! i"
if_false: fixes n :: "i32"   and tf :: "tf"   and e1s :: "b_e list"   and e2s :: "b_e list" assumes "int_eq n 0" shows "\<lparr>[$C ConstInt32 n, $b_e.If tf e1s e2s]\<rparr> \<leadsto> \<lparr>[$Block tf e2s]\<rparr>"
hcomplex_defs_10: shows "HComplex = *f2* Complex"
bvars_transaction_unfold: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" shows "bvars_transaction T = bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T) \<union> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T) \<union> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_checks T) \<union> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_updates T) \<union> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T)"
red_single_higher: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single p q f t" shows "higher q (t \<oplus> lt f) = higher p (t \<oplus> lt f)"
eq_foldr: fixes xs :: "'a list" shows "prod_list xs = foldr (*) xs (1::'a)"
sv_hom_par: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<parallel> y \<cdot> Multirelations.U = x \<cdot> Multirelations.U \<parallel> (y \<cdot> Multirelations.U)"
assms_1: shows "u \<^sup>@ k = v \<^sup>@ k"
sorted_pprt_spvec: fixes v :: "(nat \<times> 'a) list" assumes "sorted_spvec v" shows "sorted_spvec (pprt_spvec v)"
decseq_SucI: fixes X :: "nat \<Rightarrow> 'b" assumes "\<And>n. X (Suc n) \<le> X n" shows "decseq X"
sum_carrier_1: fixes f :: "nat \<Rightarrow> 'a" shows "M.sumlist (map (\<lambda>ja. f ja \<cdot>\<^sub>v gso ja) [0..<i]) \<in> carrier_vec n"
ht_size_def: fixes l :: "'a list list"   and n :: "nat" shows "ht_size l n \<equiv> n = sum_list (map length l)"
ImpR_sim: fixes G :: "'b formula"   and F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "\<^bold>\<not> G \<triangleright> F \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "\<^bold>\<not> (F \<^bold>\<rightarrow> G) \<triangleright> \<Gamma> \<turnstile> \<bottom>"
vector_sneg_minus1: fixes x :: "('c, 'd) vec" shows "- x = - (1::'c) *s x"
prod_algebra_def: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" shows "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (Pi\<^sub>E J X)) ` {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
prems: fixes x :: "'b"   and y :: "'b" assumes "x \<in> insert x__ A"   and "y \<in> insert x__ A"   and "x \<noteq> y" shows "coprime (g x) (g y)"
pcr_cr_eq: shows "pcr_complex_homo_coords = cr_complex_homo_coords"
316: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> y) \<sqinter> x = x"
lookup_tries_def: fixes t :: "('d, 'c list) trie"   and ks :: "'d list" shows "lookup_tries t ks = (case lookup_trie t ks of None \<Rightarrow> [] | Some vs \<Rightarrow> vs)"
butlast_append: fixes xs :: "'e list"   and ys :: "'e list" shows "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
Spy_analz_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> kerbV" shows "(Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
dim_cball: fixes e :: "real" assumes "0 < e" shows "dim (cball (0::'b) e) = DIM('b)"
absorb1: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "a \<sqinter> b = a"