hcju/isabelleps huggingface dataset:
simps_2: fixes n :: "nat"   and l :: "'a bdd"   and r :: "'a bdd" shows "bddh n (Branch l r) = (case n of 0 \<Rightarrow> False | Suc m \<Rightarrow> bddh m l \<and> bddh m r)"
p_rw: shows "p * p ^ (l - 1) = p ^ l"
span_self: fixes v :: "'a vec" assumes "v \<in> carrier_vec n" shows "v \<in> local.span {v}"
Radon_Nikodym: fixes N :: "'a measure" assumes "absolutely_continuous M N"   and "sets N = sets M" shows "\<exists>f\<in>borel_measurable M. density M f = N"
ord_def: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G" shows "group.ord G x \<equiv> SOME d. \<forall>n. (x [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>) = (d dvd n)"
sup2CI: fixes B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<not> B x y \<Longrightarrow> A x y" shows "sup A B x y"
simps_1: fixes uu :: "nat \<Rightarrow> nat"   and uv :: "'c list" shows "permut_aux [] uu uv = []"
group_def: fixes G :: "('b, 'c) monoid_scheme" shows "Group.group G \<equiv> Group.monoid G \<and> Group.group_axioms G"
inf_commute: fixes x :: "'e multiset"   and y :: "'e multiset" shows "x \<inter># y = y \<inter># x"
SV_inv3I: fixes x :: "'a v_state_scheme" assumes "\<And>r a b v w. votes x r a = Some v \<and> votes x r b = Some w \<Longrightarrow> v = w" shows "x \<in> SV_inv3"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "x = y"
type_equiv_def: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set" shows "t =:\<^sub>P t' \<equiv> t \<le>:\<^sub>P t' \<and> t' \<le>:\<^sub>P t"
path_inf_priorities_at_least_min_prio: fixes P :: "'a llist"   and a :: "nat" assumes "G'.valid_path P"   and "a \<in> G'.path_inf_priorities P" shows "Min (\<omega>\<^bsub>G'\<^esub> ` V\<^bsub>G'\<^esub>) \<le> a"
elims_3: fixes x :: "atom"   and xa :: "'b list" assumes "\<not> I\<^sub>d\<^sub>l\<^sub>o x xa" obtains i :: "nat" and j :: "nat" and xs :: "'b list" where "x = Eq i__ j__" and "xa = xsa__" and "xsa__ ! i__ \<noteq> xsa__ ! j__"   | i :: "nat" and j :: "nat" and xs :: "'b list" where "x = Less i j" and "xa = xs" and "\<not> xs ! i < xs ! j"
circuit_def: fixes X :: "'a set" shows "circuit X = (X \<subseteq> carrier \<and> \<not> indep X \<and> (\<forall>x\<in>X. indep (X - {x})))"
assms_1: shows "g \<in> M \<rightarrow>\<^sub>M M"
assms_3: shows "set_borel_measurable borel {A<..<B} f"
ON: shows "(+) \<alpha> ` elts \<beta> \<subseteq> ON"
line_integral_exists_on_typeI_Cube_boundaries': fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "green_typeI_cube i j twoC F"   and "(k, \<gamma>) \<in> boundary twoC" shows "line_integral_exists F {i} \<gamma>"
graph_def: fixes G :: "('a, 'b) graph_rec_scheme" shows "graph G \<equiv> g_V0 G \<subseteq> g_V G \<and> g_E G \<subseteq> g_V G \<times> g_V G"
unat_eq_zero: fixes x :: "'b word" shows "(unat x = 0) = (x = 0)"
Im3_def: fixes quat :: "quat" shows "Im3 quat = (case quat of Quat x1 x2 x3 x4 \<Rightarrow> x4)"
subreachableI: fixes A :: "(('e \<Rightarrow> 'f) \<times> 'g, 'h) automaton"   and J :: "'e set"   and U :: "('e \<Rightarrow> 'f) \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool" assumes "local_steps (automaton.trans A) J"   and "other_steps U J" shows "subreachable A U J"
1: shows "w\<^sub>2 =\<^sub>F w\<^sub>1"
renumber: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and r :: "('a \<times> 'a) set" assumes "\<forall>i. (a, f i) \<in> r\<^sup>* \<and> (f i, f (Suc i)) \<in> r" shows "\<exists>f. f 0 = a \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)"
ardens_lemma_equiv: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "\<not> ewp x" shows "(z + x \<cdot> y = y) = (x\<^sup>\<star> \<cdot> z = y)"
rcosets_part_G: fixes H :: "'a set" assumes "subgroup H G" shows "\<Union> (rcosets H) = carrier G"
simps_1: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "ring.eval R [] = (\<lambda>_. \<zero>\<^bsub>R\<^esub>)"
modus_ponens: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]"   and "[\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<psi> in v]"
intros: fixes p :: "'a \<times> 'b" shows "fst p \<in> Basic_BNFs.fsts p"
assms_8: fixes x :: "'b \<times> 'c" assumes "x \<in> space (S \<Otimes>\<^sub>M T)" shows "0 \<le> Pxy x"
B1_def: shows "B1 = {LEAST x. x \<in> U |U. U \<in> A}"
approx_refl: fixes x :: "'c star" shows "x \<approx> x"
continuous_on_norm_id: fixes S :: "'a set" shows "continuous_on S norm"
union_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "F g (A \<union> B) = F g A \<^bold>* F g B"
dvd_zlcms: fixes i :: "int"   and "is" :: "int list" assumes "i \<in> set is" shows "i dvd zlcms is"
trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst'': fixes t :: "('a, 'b) Term.term"   and b :: "('a, 'b) stateful_strand_step"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)" shows "\<exists>s\<in>trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p b. t = s \<cdot> rm_vars (set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p b)) \<delta>"
M_dim: shows "M \<in> carrier_mat (nr + 1) (n + 1)"
UPD_STATE_def: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int"   and x :: "char list"   and v :: "int \<Rightarrow> int" shows "UPD_STATE s x v \<equiv> s(x := v)"
sim: shows "\<forall>P Q P'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q')"
normal_inv_iff: fixes N :: "'a set set" shows "N \<lhd> G Mod N = (subgroup N (G Mod N) \<and> (\<forall>x\<in>carrier (G Mod N). \<forall>h\<in>N. x \<otimes>\<^bsub>G Mod N\<^esub> h \<otimes>\<^bsub>G Mod N\<^esub> inv\<^bsub>G Mod N\<^esub> x \<in> N))"
concat_take_suc: fixes j :: "nat"   and ws :: "'b list list" assumes "j < \<^bold>|ws\<^bold>|" shows "concat (take j ws) \<cdot> ws ! j = concat (take (Suc j) ws)"
distinct_13: shows "hp_IdleInit \<noteq> hp_IdleMarkSweep"
inv_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<otimes>\<^bsub>H'\<^esub> y = \<one>\<^bsub>H'\<^esub>"   and "x \<in> carrier H'"   and "y \<in> carrier H'" shows "y \<otimes>\<^bsub>H'\<^esub> x = \<one>\<^bsub>H'\<^esub>"
order_def: fixes a :: "'b G"   and b :: "'b G" shows "a \<le> b \<equiv> a l\<rightarrow> b = 1"
uncommitted_reads_see_hb_def: fixes P :: "'a prog"   and J :: "nat \<Rightarrow> ('b, 'c) justifying_execution" shows "uncommitted_reads_see_hb P J = (\<forall>n. \<forall>r'\<in>read_actions (justifying_exec (J (Suc n))). action_translation (J (Suc n)) r' \<in> action_translation (J n) ` committed (J n) \<or> P,justifying_exec (J (Suc n)) \<turnstile> justifying_ws (J (Suc n)) r' \<le>hb r')"
OFS_def: fixes a :: "'p"   and b :: "'p"   and c :: "'p"   and d :: "'p"   and a' :: "'p"   and b' :: "'p"   and c' :: "'p"   and d' :: "'p" shows "OFS a b c d a' b' c' d' \<equiv> B a b c \<and> B a' b' c' \<and> a b \<equiv> a' b' \<and> b c \<equiv> b' c' \<and> a d \<equiv> a' d' \<and> b d \<equiv> b' d'"
is_preserved_by_base_iso: fixes \<phi> :: "'a"   and r' :: "'a" assumes "\<guillemotleft>\<phi> : r'__ \<Rightarrow> r'\<guillemotright>"   and "local.iso \<phi>" shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r' ((\<phi> \<star> f) \<cdot> \<rho>') f g"
seqstar_inf_parstar: fixes a :: "'c" shows "a\<^sup>; \<le>\<^sub>\<K> a\<^sup>*"
respects_unit: fixes a :: "'c" assumes "C.obj a" shows "(\<rho>\<^sub>0 a \<star>\<^sub>D G.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<rho>\<^sub>0 a] = \<rho>\<^sub>1 a \<cdot>\<^sub>D (H.unit a \<star>\<^sub>D \<rho>\<^sub>0 a)"
has_limits_of_shape_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "D.J.has_limits_of_shape J \<equiv> \<forall>D. diagram J J__ D \<longrightarrow> (\<exists>a \<chi>. limit_cone J J__ D a \<chi>)"
of_nat_closed: fixes R :: "('b, 'c) ring_scheme"   and n :: "nat" assumes "ring R" shows "\<guillemotleft>n\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> \<in> carrier R"
assms_4: fixes i :: "nat"   and x :: "real" assumes "i < length ps - 2"   and "poly (ps ! (i + 1)) x = 0" shows "sgn (poly (ps ! (i + 2)) x) = - sgn (poly (ps ! i) x)"
pbernpoly_bigo: fixes n :: "nat" shows "pbernpoly n \<in> O(\<lambda>_. 1)"
Icarr: fixes i :: "'a" assumes "i \<in> a_kernel R S h" shows "i \<in> carrier R"
fg_round_numbers_increase: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and n :: "nat"   and m :: "nat"   and p :: "'f" assumes "fg_run A rho HOs SHOs coords"   and "n \<le> m" shows "round (rho n) p \<le> round (rho m) p"
of_nat_binomial_Suc: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "of_nat (Suc n choose k) = of_nat (Suc n) / of_nat (Suc n - k) * of_nat (n choose k)"
ux: shows "\<forall>w. P w (u \<ominus> x) = (P w u \<and> \<not> O w x)"
select_convs_1: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'b" shows "prg \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = prg"
ArrayClear: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (CLEAR x[], s) \<Rightarrow> s(x := \<lambda>_. 0)"
Rep_Abs_Bot: shows "Rep_process (Abs_process ({(s, X). front_tickFree s}, {d. front_tickFree d})) = ({(s, X). front_tickFree s}, {d. front_tickFree d})"
cod_simps_2: shows "\<mu>\<nu>.cod.leg0 = \<mu>\<nu>.leg0"
e: fixes u :: "'a" assumes "a = b \<otimes> u"   and "u \<in> Units G" shows "P"
some_trg_composable: fixes \<sigma> :: "'a"   and \<mu> :: "'a" assumes "arr \<sigma>" shows "(\<mu> \<star> \<sigma> \<noteq> null) = (\<mu> \<star> some_trg \<sigma> \<noteq> null)"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" assumes "indRelRp a1 a2" obtains S :: "'procS" where "S \<in>S a1" and "\<lbrakk>S\<rbrakk> \<in>T a2"
set_integral_Un: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "A \<inter> B = {}"   and "set_integrable M A f"   and "set_integrable M B f" shows "set_lebesgue_integral M (A \<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f"
LFO_pre_refine: fixes li :: "'a list"   and l :: "'b set"   and A :: "('a \<times> 'b) set"   and ci :: "'c \<Rightarrow> bool"   and c :: "'d \<Rightarrow> bool"   and R :: "('c \<times> 'd) set"   and fi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c nres"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'd nres"   and s0i :: "'c"   and s0 :: "'d"   and I :: "'b set \<Rightarrow> 'd \<Rightarrow> bool" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "(ci, c) \<in> R \<rightarrow> bool_rel"   and "(fi, f) \<in> A \<rightarrow> R \<rightarrow> \<langle>R\<rangle>nres_rel"   and "(s0i, s0) \<in> R" shows "LIST_FOREACH' (RETURN li) ci fi s0i \<le> \<Down> R (FOREACH\<^sub>C\<^bsup>I\<^esup> l c f s0)"
lunit_char_2: fixes f :: "'a" assumes "ide f" shows "L \<l>[f] = (\<i>[trg f] \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[trg f, trg f, f]"
lcm_mult_left: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "lcm (times c a) (times c b) = normalize (times c (lcm a b))"
Quotient_finfun: shows "Quotient (eq_onp (\<lambda>x. x \<in> finfun)) Abs_finfun finfun_apply cr_finfun"
mono_ndet_FD: fixes P :: "'b process"   and P' :: "'b process"   and S :: "'b process"   and S' :: "'b process" assumes "P \<le> P'"   and "S \<le> S'" shows "(P |-| S) \<le> (P' |-| S')"
Suc_innermost_id1': fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and j' :: "nat"   and i' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "j' \<le> n"   and "i' < i" shows "fw m n (Suc k) i' j' i j = fw m n k n n i j"
rec_pdec2_def: shows "rec_pdec2 = CN rec_minus [CN rec_max_triangle [recf.Id 1 0], CN rec_pdec1 [recf.Id 1 0]]"
simps_2: fixes t :: "'a Trace"   and s :: "'a" shows "tFirst (t \<leadsto> s) = tFirst t"
image_of_inverse: fixes x :: "'a" assumes "x \<in> H" shows "x \<in> inverse ` H"
simps_2: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and C' :: "char list"   and Cs :: "char list list" shows "exec_step_input P C M pc (Called (C' # Cs)) = StepC2 C' Cs"
l0: shows "leaves i l \<inter> leaves (k + 1) r = {}"
poly_pCons: fixes a :: "'b"   and p :: "'b poly"   and x :: "'b" shows "poly (pCons a p) x = a + x * poly p x"
moebius_mat_eq_refl: fixes x :: "moebius_mat" shows "moebius_mat_eq x x"
face_distance_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and F :: "'b set" assumes "ChamberComplexIsomorphism X Y f"   and "SimplicialComplex.maxsimp X C"   and "F \<in> X" shows "ChamberComplex.face_distance Y (f ` F) (f ` C) = ChamberComplex.face_distance X F C"
assms_2: shows "S t i \<turnstile> RecvMarker cid p q \<mapsto> S t (i + 1)"
1: shows "local.inv g = f \<cdot> local.inv (g \<cdot> f)"
sints_def: shows "sints 32 = range (signed_take_bit (32 - 1))"
arg_uminus: fixes z :: "complex" assumes "z \<noteq> 0" shows "arg (- z) = \<downharpoonright>arg z + pi\<downharpoonleft>"
\<Psi>_sel_3: shows "source \<Psi> = SOURCE"
verit_comp_simplify1_3: fixes b' :: "'f"   and a' :: "'f" shows "(\<not> b' \<le> a') = (a' < b')"
InterpEqE: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and e2 :: "('a, 'b) Expression"   and B :: "'a"   and i :: "('c, 'd) IType" assumes "L\<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i" obtains g1 :: "'d" and g2 :: "'d" where "i = IBool (g1 = g2)" and "L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g1" and "L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g2"
disjuncts_qf: fixes p :: "fm" assumes "qfree p" shows "\<forall>q\<in>set (disjuncts p). qfree q"
delete_update_1: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.delete k (Mapping.update k v m) = Mapping.delete k m"
p_cost: shows "\<mu> c s u = ereal (awalk_cost c p)"
diff_1: shows "integrable M (\<lambda>x. u x - f 0 x)"
Always_ConstrainsD: fixes F :: "'b program"   and INV :: "'b set"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> Always INV"   and "F \<in> A Co A'" shows "F \<in> A Co INV \<inter> A'"
vector_all_zero_from_index_eq: fixes i :: "'d"   and A :: "('e, 'd) vec" shows "(\<forall>m\<ge>i. A $ m = (0::'e)) = vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray A)"
lists_succ_permutes: fixes xss :: "'b list list" assumes "distincts xss" shows "lists_succ xss permutes \<Union> (sset xss)"
standard_open_is_subset: fixes x :: "('a \<times> 'a) set" assumes "x \<in> local.carrier_quotient_ring" shows "local.local.standard_open x \<subseteq> local.local.spectrum"
map_S_def: shows "map_S = map_fun id (map_fun id (map_fun id (map_fun id (map_fun id (map_fun id (map_fun Rep_S Abs_S)))))) map_G"
induct: fixes P :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "nat"   and a2 :: "'a list" assumes "\<And>acc n. P acc n []"   and "\<And>acc n x xs. (\<And>nat. n = Suc nat \<Longrightarrow> P (x # acc) nat xs) \<Longrightarrow> P acc n (x # xs)" shows "P a0 a1 a2"
ys_p: fixes i :: "nat" assumes "i < n" shows "ys ! i = p i"
iterative_def: fixes S :: "'d set"   and x :: "'d" shows "iterative S x \<equiv> x \<notin> S"
sl: shows "\<Turnstile> mset \<Gamma>' \<Rightarrow> mset \<Delta>'"
keys_minus_higher: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "keys (p - higher p v) = {u \<in> keys p. u \<preceq>\<^sub>t v}"
product_in_first_summand: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "\<not> O y z" shows "P ((x \<oplus> y) \<otimes> (x \<oplus> z)) x"
in_prime_factors_imp_prime: fixes p :: "'b poly"   and x :: "'b poly" assumes "p \<in># field_poly.prime_factorization x" shows "field_poly.prime p"
str_eq_def: fixes A :: "'c list set" shows "\<approx>A \<equiv> {(x, y). \<forall>z. (x @ z \<in> A) = (y @ z \<in> A)}"
ring: shows "ring (R\<lparr>carrier := K\<rparr>)"
poly_mult_is_polynomial: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p1 :: "'c list"   and p2 :: "'c list" assumes "ring R"   and "subring K R"   and "set p1 \<subseteq> K"   and "set p2 \<subseteq> K" shows "polynomial\<^bsub>R\<^esub> K (ring.poly_mult R p1 p2)"
comp_elim: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and Q :: "'d \<times> 'e \<Rightarrow> bool"   and ac :: "'c \<times> 'e" assumes "(P ;; Q) ac" obtains a :: "'c" and b :: "'d" and c :: "'e" where "ac = (a, c)" and "P (a, b)" and "Q (b, c)"
perzeta_def: fixes q' :: "real"   and s :: "complex" shows "perzeta q' s = (if q' \<in> \<int> then zeta s else let q = frac q' in if s = 0 then \<i> / complex_of_real (2 * pi) * (pre_zeta q 1 - pre_zeta (1 - q) 1 + Ln (1 - complex_of_real q) - Ln (complex_of_real q) + complex_of_real pi * \<i>) else if s \<in> \<nat> then eval_fds (fds_perzeta q) s else complex_of_real (2 * pi) powr (s - 1) * \<i> * Gamma (1 - s) * (\<i> powr - s * hurwitz_zeta q (1 - s) - \<i> powr s * hurwitz_zeta (1 - q) (1 - s)))"
map_add_restrict_comp_left: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "m |` (UNIV - S) ++ m |` S = m"
less_name_code_2: fixes x :: "char"   and xs :: "char list" shows "(Name [] < Name (x # xs)) = True"
iMOD_in_i_set_mult_imp_divisor_mod_0: fixes m :: "nat"   and r :: "nat"   and k :: "nat" assumes "m \<noteq> Suc 0"   and "[ r, mod m ] \<in> i_set_mult k" shows "m mod k = 0"
app3_list_lem1: fixes sf1 :: "('c, 'd) Digit"   and ts :: "('c, 'd) Node list"   and pr2 :: "('c, 'd) Digit" shows "nlistToList (nodes (digitToNlist sf1 @ ts @ digitToNlist pr2)) = digitToList sf1 @ nlistToList ts @ digitToList pr2"
FrameStatImp_def: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and G :: "'d frame" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.FrameStatImp SImp F G \<equiv> \<forall>\<phi>. assertionAux.FrameImp SImp F \<phi> \<longrightarrow> assertionAux.FrameImp SImp G \<phi>"
cases: fixes x :: "'b list \<times> 'b list" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)"
that: fixes t' :: "'x \<Rightarrow>\<^sub>0 nat"   and x :: "'x" assumes "t' \<in> .[X]"   and "x \<in> X"   and "V \<subseteq> U"   and "0 \<notin> (\<lambda>s. s - t') ` S"   and "s = t' + t + monomial 1 x"   and "(monomial (1::'a) s, V) \<in> set (fst (split (t' + t) V ((\<lambda>s. s - t') ` S)))"   and "set (snd (split (t' + t) V ((\<lambda>s. s - t') ` S))) \<subseteq> set (snd (split t U S))" shows "thesis"
assms_2: fixes p :: "nat"   and i :: "nat" assumes "prime p"   and "0 < i" shows "f (p ^ i) = g p i"
finite_nlists: fixes A :: "'c set"   and n :: "nat" assumes "finite A" shows "finite (nlists A n)"
extractableE: fixes X :: "msg"   and K :: "msg set"   and I :: "msg set" assumes "X \<in> extractable K I" obtains A :: "agent" and B :: "agent" and M :: "msg" where "X = implInsec Aa__ Ba__ Ma__" and "X \<in> I"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implAuth Ab__ Bb__ Mb__" and "X \<in> I"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implConfid Ac__ Bc__ Mc__" and "X \<in> I" and "(Ac__, Bc__) \<in> broken K"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implSecure A B M" and "X \<in> I" and "(A, B) \<in> broken K"
f_expand_append: fixes xs :: "'d message_af list"   and ys :: "'d message_af list"   and k :: "nat" shows "(xs @ ys) \<odot> k = xs \<odot> k @ ys \<odot> k"
nilTrans_3: fixes \<Psi> :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> \<tau> \<prec> P'" shows "False"
is_denormal_denormal_of_Float: fixes x :: "Float.float" assumes "is_denormal_Float TYPE(('a, 'b) IEEE.float) x" shows "is_denormal (denormal_of_Float x)"
ls_pick_hnr: assumes "Sepref_Constraints.CONSTRAINT is_pure A" shows "(return \<circ> hd, mop_set_pick) \<in> local.list_set_assn\<^sup>k \<rightarrow>\<^sub>a A"
lipschitz_on_cong: fixes C :: "real"   and D :: "real"   and U :: "'d set"   and V :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" assumes "C = D"   and "U = V"   and "\<And>x. x \<in> V \<Longrightarrow> g x = f x" shows "C-lipschitz_on U g = D-lipschitz_on V f"
AssertionStatImpTrans: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" assumes "\<Psi> \<hookrightarrow> \<Psi>'"   and "\<Psi>' \<hookrightarrow> \<Psi>''" shows "\<Psi> \<hookrightarrow> \<Psi>''"
not_fin: shows "\<forall>i<k. \<not> final (head (f i))"
nth_init: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "i < Suc n" shows "e_nth (f \<triangleright> n) i = the (f i)"
distinct_99: fixes x81 :: "'a ltln"   and x82 :: "'a ltln"   and x91 :: "'a ltln"   and x92 :: "'a ltln" shows "x81 U\<^sub>n x82 \<noteq> x91 R\<^sub>n x92"
RestrictedP_HDomain_Incl: fixes s :: "tm"   and k :: "tm"   and s' :: "tm" shows "{HDomain_Incl s k, RestrictedP s k s'} \<turnstile> HDomain_Incl s' k"
insert_uconnectedV_per: fixes x :: "'a"   and y :: "'a"   and F :: "'a uprod set" assumes "x \<noteq> y"   and "x \<in> verts"   and "y \<in> verts"   and "F \<subseteq> E" shows "Restr (uconnected (insert (Upair x y) F)) verts = per_union (Restr (uconnected F) verts) x y"
point_rep_number_inv: fixes x :: "'a" shows "int (size (filter_mset ((\<notin>) x) \<B>)) = \<b> - \<B> rep x"
epath_restrict': fixes e :: "'b uprod"   and E :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list"   and v :: "'b" assumes "epath (insert e E) u p v"   and "e \<notin> set p" shows "epath E u p v"
\<alpha>_limit_set_eq_rev: fixes x :: "'a" shows "\<alpha>_limit_set x = rev.\<omega>_limit_set x"
simps_3: fixes F :: "'b set \<Rightarrow> 'a"   and c1 :: "com"   and c2 :: "com"   and M :: "'b acom set" shows "lift F (c1;; c2) M = lift F c1 (sub\<^sub>1 ` M);; lift F c2 (sub\<^sub>2 ` M)"
RA3: fixes P :: "('b, 'c \<times> 'd) uexpr" shows "P\<^sup>-\<^sup>- = P"
less_le_not_le: fixes x :: "'a word"   and y :: "'a word" shows "(x <s y) = (x \<le>s y \<and> \<not> y \<le>s x)"
smc_Obj_if_Cod_vrange: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>)" shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
coincidence_formula: fixes \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and \<phi> :: "fml"   and I :: "interp" assumes "Vagree \<omega> \<omega>' (FVF \<phi>)" shows "(\<omega> \<in> fml_sem I \<phi>) = (\<omega>' \<in> fml_sem I \<phi>)"
CatExp_def: fixes A :: "('g, 'i, 'k) Category_scheme"   and B :: "('h, 'j, 'l) Category_scheme" shows "CatExp A B \<equiv> MakeCat (CatExp' A B)"
isCont_tanh: fixes x :: "'a" assumes "cosh x \<noteq> (0::'a)" shows "isCont tanh x"