hcju/isabelleps huggingface dataset:
And_prop_entailment: fixes S :: "'a ltl set"   and xs :: "'a ltl list" shows "S \<Turnstile>\<^sub>P And xs = (\<forall>x\<in>set xs. S \<Turnstile>\<^sub>P x)"
assoc_in_homs_2: shows "\<guillemotleft>\<lparr>Chn = \<mu>_\<nu>\<pi>_\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<mu> \<star> (\<nu> \<star> \<pi>) \<star> \<rho>)\<rparr> : (\<mu> \<star> \<nu> \<star> \<pi>) \<star> \<rho> \<Rightarrow> \<mu> \<star> (\<nu> \<star> \<pi>) \<star> \<rho>\<guillemotright>"
Sup_cl_op_im_def: shows "Sup \<equiv> map_fun ((`) Rep_cl_op_im) Abs_cl_op_im (\<lambda>X. cl_op (\<Squnion> X))"
Derives1_append_suffix: fixes v :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and w :: "symbol list"   and u :: "symbol list" assumes "Derives1 v i r w"   and "is_sentence u" shows "Derives1 (v @ u) i r (w @ u)"
power_inductl: fixes z :: "('a \<times> ('b \<times> 'a) list) set"   and x :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set"   and n :: "nat" assumes "z \<union> t_prod x y \<subseteq> y" shows "t_prod (trace_dioid.power x n) z \<subseteq> y"
nprv_hyp: fixes \<phi> :: "'fmla"   and F :: "'fmla set" assumes "\<phi> \<in> F"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F \<phi>"
poly_p: shows "poly p (Fraction_Field.Fract a b) = Fraction_Field.Fract q0 (b ^ m0)"
sequential_interchange: fixes c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d\<^sub>0 :: "'b::refinement_lattice"   and d\<^sub>1 :: "'b::refinement_lattice" shows "(c\<^sub>0::'b::refinement_lattice) ; (c\<^sub>1::'b::refinement_lattice) \<iinter> (d\<^sub>0::'b::refinement_lattice) ; (d\<^sub>1::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>0 \<iinter> d\<^sub>0) ; (c\<^sub>1 \<iinter> d\<^sub>1)"
token_succeeds_run_merge: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "token_run x n = token_run y n"   and "token_succeeds x" shows "token_succeeds y"
monom_vars_def: fixes m :: "'c monom" shows "monom_vars m = set (monom_vars_list m)"
dgrad_p_set_le_trans: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and G :: "('t \<Rightarrow>\<^sub>0 'e) set"   and H :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "dgrad_p_set_le d F G"   and "dgrad_p_set_le d G H" shows "dgrad_p_set_le d F H"
that: fixes s :: "'t" assumes "s \<le> t \<and> 0 < zcount (c_imp c loc + c_work c loc) s" shows "thesis"
assms_2: shows "\<forall>t t'. P t \<longrightarrow> t' \<sqsubseteq> t \<longrightarrow> P t'"
Lim_at: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and a :: "'a" shows "f \<midarrow>a\<rightarrow> l = (\<forall>e>0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
not_in_sum_var: fixes var :: "nat"   and f :: "nat \<Rightarrow> real mpoly"   and n :: "nat" assumes "\<forall>i. var \<notin> vars (f i)" shows "var \<notin> vars (sum f {0..<n})"
rotate_pow_self: fixes l :: "nat"   and u :: "'b list"   and k :: "nat" shows "rotate (l * \<^bold>|u\<^bold>|) (u \<^sup>@ k) = u \<^sup>@ k"
d2: fixes x :: "'a" assumes "x \<in> {a--b}" shows "0 < d2 x"
simps: fixes n :: "nat" shows "D n = C n - pderiv (B n)"
comp_ndet_l_distr: fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and Q :: "'c \<times> 'c \<Rightarrow> bool"   and R :: "'c \<times> 'c \<Rightarrow> bool" shows "(P \<or> Q ;; R) = ((P ;; R) \<or> (Q ;; R))"
extendRule_def: fixes forms :: "'a sequent"   and R :: "'a sequent list \<times> 'a sequent" shows "extendRule forms R \<equiv> (map (extend forms) (fst R), extend forms (snd R))"
sets_measure_pmf_count_space: fixes M :: "'e pmf" shows "measure_pmf.events M = sets (count_space UNIV)"
dist_commute: fixes dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.metric_space dist uniformity open" shows "dist x y = dist y x"
sub1: fixes C :: "'a set" assumes "C \<in> \<F>'" shows "\<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
derives_implies_Derivation: fixes a :: "symbol list"   and b :: "symbol list" assumes "derives a b" shows "\<exists>D. Derivation a D b"
act_act: fixes g :: "'a"   and h :: "'a"   and b :: "'b" assumes "g \<in> carrier G"   and "h \<in> carrier G" shows "b <o g <o h = b <o g \<otimes> h"
outstanding_volatile_writes_unowned_by_others_instructions_read_value_independent: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "outstanding_volatile_writes_unowned_by_others ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "outstanding_volatile_writes_unowned_by_others (ts[i := (p', is', \<theta>', sb, \<D>', \<O>, \<R>')])"
inner_not_same_Basis: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and u :: "'a"   and v :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis"   and "u \<noteq> v" shows "inner u v = 0"
remdups_id_iff_distinct: fixes xs :: "'g list" shows "(remdups xs = xs) = distinct xs"
inf_assoc: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<inter># y \<inter># z = x \<inter># (y \<inter># z)"
face_set_eq: shows "\<F> g' \<union> {f} = {f12, f21} \<union> \<F> g"
isomorphism_def: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" shows "ciso\<^bsub>C\<^esub> f \<equiv> \<exists>g. cinv\<^bsub>C\<^esub> f g"
hyps_3: shows "finite {M. vars (elements M) = V' \<and> uniq (elements M) \<and> consistent (elements M)}"
extension_onD: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "extension_on A r s"   and "x \<in> A"   and "y \<in> A"   and "(x, y) \<in> s" shows "(x, y) \<in> r"
diff_add_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "plus (minus a b) c = minus (plus a c) b"
hf: fixes x :: "'a" assumes "x \<in> \<Union> \<G>" shows "h x = f x"
lfilter_lfilter: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lfilter P (lfilter Q xs) = lfilter (\<lambda>x. P x \<and> Q x) xs"
subformulas\<^sub>\<mu>_finite: fixes \<phi> :: "'b ltln" shows "finite (subformulas\<^sub>\<mu> \<phi>)"
simps_3: fixes h1 :: "'b hp"   and h2 :: "'b hp" shows "merge (Some h1) (Some h2) = Some (link h1 h2)"
valid_path: shows "\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>"
rank_invar_meld_strong: fixes bq1 :: "('c, 'd) BinomialTree list"   and bq2 :: "('c, 'd) BinomialTree list" assumes "rank_invar bq1"   and "rank_invar bq2" shows "rank_invar (meld bq1 bq2) \<and> min (rank (hd bq1)) (rank (hd bq2)) \<le> rank (hd (meld bq1 bq2))"
lemma_4_2_i: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a l\<rightarrow> b = a * c l\<rightarrow> b * c"
of_complex_zero: shows "of_complex 0 = 0\<^sub>h"
blocks_old_induct: fixes P :: "char list list \<Rightarrow> ty list \<Rightarrow> val list \<Rightarrow> expr \<Rightarrow> bool"   and u :: "char list list"   and v :: "ty list"   and w :: "val list"   and x :: "expr" assumes "\<And>aj ak al. P [] [] (aj # ak) al"   and "\<And>ad ae a b. P [] (ad # ae) a b"   and "\<And>V Vs a b. P (V # Vs) [] a b"   and "\<And>V Vs T Ts aw. P (V # Vs) (T # Ts) [] aw"   and "\<And>V Vs T Ts v vs e. P Vs Ts vs e \<Longrightarrow> P (V # Vs) (T # Ts) (v # vs) e"   and "\<And>e. P [] [] [] e" shows "P u v w x"
quadratic_eq_factoring: fixes D :: "real"   and b :: "real"   and a :: "real"   and c :: "real"   and x\<^sub>1 :: "real"   and x\<^sub>2 :: "real"   and x :: "real" assumes "D = b\<^sup>2 - 4 * a * c"   and "0 \<le> D"   and "x\<^sub>1 = (- b + sqrt D) / (2 * a)"   and "x\<^sub>2 = (- b - sqrt D) / (2 * a)"   and "a \<noteq> 0" shows "a * x\<^sup>2 + b * x + c = a * (x - x\<^sub>1) * (x - x\<^sub>2)"
R28_d: shows "pmf (sds R28) d = 1 - pmf (sds R28) a"
pw_leI': fixes S' :: "'c nres"   and S :: "'c nres" assumes "nofail S' \<Longrightarrow> nofail S"   and "\<And>x. \<lbrakk>nofail S'; inres S x\<rbrakk> \<Longrightarrow> inres S' x" shows "S \<le> S'"
h12_def: shows "h12 = (\<lambda>t. Im ((g1 +++ g2) t - z) / Re ((g1 +++ g2) t - z))"
verit_negate_coefficient_3: fixes a :: "'a"   and b :: "'a" assumes "a = b" shows "- a = - b"
sum_list_subtractf: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.ab_group_add plus zero minus uminus" shows "monoid_add.sum_list plus zero (map (\<lambda>x. minus (f x) (g x)) xs) = minus (monoid_add.sum_list plus zero (map f xs)) (monoid_add.sum_list plus zero (map g xs))"
card_of_refl: fixes A :: "'a set" shows "ordIso2 (card_of A) (card_of A)"
bl2wc_exp_ex: fixes c :: "cell list"   and m :: "nat" assumes "Suc (bl2wc c) = 2 ^ m" shows "\<exists>rs n. c = Oc \<up> rs @ Bk \<up> n"
var\<^sub>A_of_def: shows "var\<^sub>A_of \<equiv> inv var\<^sub>C_of"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> set_cr" shows "\<exists>A. Says A B X \<in> set evs"
Labeling_axioms_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "Labeling_axioms c s t f l \<equiv> (\<forall>u v. (u, v) \<in> Graph.E (residualGraph c f) \<longrightarrow> l u \<le> l v + 1) \<and> l s = card (Graph.V c) \<and> l t = 0"
clock_jview_eq_inv: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envObs :: "'b \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'b"   and t' :: "'f Trace"   and t :: "'f Trace" assumes "Environment jkbp" shows "(Environment.clock_jview envObs a t' = Environment.clock_jview envObs a t) = (tLength t' = tLength t \<and> envObs a (tLast t') = envObs a (tLast t))"
lub_fun: fixes S :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "chain S" shows "(\<Squnion>i. S i) = (\<lambda>x. \<Squnion>i. S i x)"
commutativities_1: shows "\<mu>.cod.leg0 \<cdot> \<mu>.chine \<cdot> doms.Prj\<^sub>1\<^sub>1 = \<nu>.cod.leg1 \<cdot> \<nu>.chine \<cdot> doms.Prj\<^sub>0\<^sub>1"
assms_3: shows "\<forall>e\<in>pred c. finite (Bexp.vars e)"
T_BIT_def: fixes n :: "nat" shows "T_BIT n = sum t_BIT {..<n}"
simps_1: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_graph G" shows "valid_graph.is_path G v [] v' = (v = v' \<and> v' \<in> nodes G)"
coprime_common_divisor: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "algebraic_semidom.coprime times one a b"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "dvd.dvd times c one"
COND_refine: fixes wl :: "'V \<Rightarrow> 'W infty option"   and res :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and it :: "('W \<times> 'V) set"   and wl' :: "'V \<Rightarrow> 'W infty option"   and res' :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and w' :: "'W"   and v'a :: "'V" assumes "uinvarm v (\<alpha>w wl) (\<alpha>r res) it (wl', res')"   and "dinvarm (wl, res)" shows "(mpath_weight' (res v) + Num w' < mpath_weight' (res' v'a)) = (path_weight' (\<alpha>r res v) + Num w' < path_weight' (\<alpha>r res' v'a))"
subdistl_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.ab_pre_semiring plus times" shows "plus (times z x) (times z (plus x y)) = times z (plus x y)"
sorted_list_of_pdevs_domain_eq: fixes X :: "'b pdevs" shows "sorted_list_of_set (pdevs_domain X) = filter ((\<noteq>) (0::'b) \<circ> pdevs_apply X) [0..<degree X]"
intro: fixes \<Delta> :: "('a, 'b) network_scheme" assumes "countable_network \<Delta>" shows "antiparallel_edges \<Delta>"
has_contour_integral_mirror_iff: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex"   and I :: "complex" assumes "valid_path g" shows "(f has_contour_integral I) (- g) = ((\<lambda>x. - f (- x)) has_contour_integral I) g"
nat_uint_eq: fixes w :: "'a word" shows "nat (uint w) = unat w"
primes_theta_def: shows "\<theta> = prime_sum_upto (\<lambda>p. ln (real p))"
group_commutes_pow: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x [^] n \<otimes> y = y \<otimes> x [^] n"
nonneg: shows "nonneg_cexpr (shift_var_set (set vs \<union> set vs')) (case_nat BOOL \<Gamma>) f"
subst_idemE: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "subst_idem \<sigma>" shows "subst_domain \<sigma> \<inter> range_vars \<sigma> = {}"
cases: fixes a :: "Vertices list" assumes "walk a" obtains "a = []"   | v :: "Vertices" where "a = [v__]" and "v__ \<in> V\<^bsub>T2\<^esub>"   | v :: "Vertices" and w :: "Vertices" and vs :: "Vertices list" where "a = v # w # vs" and "v \<rightarrow>\<^bsub>T2\<^esub> w" and "walk (w # vs)"
chf_lm_5: fixes A :: "nat set" assumes "finite A" shows "chf A \<in> PrimRec1"
add_block_wf_cond: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection"   and "b \<subseteq> point_set" shows "incidence_system point_set (incidence_system.add_block block_collection b)"
aligned_add_offset_less: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word"   and z :: "'b word" assumes "is_aligned x n"   and "is_aligned y n"   and "x < y"   and "z < 2 ^ n" shows "x + z < y"
assms_2: shows "(x, b) \<in> closed_segment (x0, y0) (x1, y1)"
gallery_chamber_system: fixes Cs :: "'a set list" assumes "folding.gallery Cs" shows "order.greater_eq folding.\<C> (set Cs)"
fmfilter_upd: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and m :: "('a, 'b) fmap" shows "fmfilter P (fmupd x y m) = (if P x then fmupd x y (fmfilter P m) else fmfilter P m)"
ennreal_def: shows "ennreal \<equiv> map_fun id e2ennreal (sup 0 \<circ> ereal)"
intersects_x_axis_positive_intersects_y_axis_positive: fixes H :: "circline" shows "intersects_x_axis_positive (moebius_circline (moebius_rotation (- pi / 2)) H) = intersects_y_axis_positive H"
is_arg_minD1: fixes m :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "is_arg_min m P x" shows "P x"
Un_ac_3: fixes A :: "'e set"   and B :: "'e set" shows "A \<union> B = B \<union> A"
rep_eq: fixes x :: "'b set_rbt" shows "keys x = RBT_Impl.keys (mapping_rbt.impl_of x)"
uint32_set_bit_def: fixes x :: "uint32"   and n :: "integer"   and b :: "bool" shows "uint32_set_bit x n b = (if n < 0 \<or> 31 < n then undefined set_bit_class.set_bit x n b else set_bit_class.set_bit x (nat_of_integer n) b)"
simps_1: fixes P :: "'g \<Rightarrow> bool" shows "dropWhile P [] = []"
eventually_nhds_iff_sequentially: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "class.first_countable_topology open" shows "eventually P (topological_space.nhds open a) = (\<forall>f. filterlim f (topological_space.nhds open a) sequentially \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n)))"
ImplI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] \<Longrightarrow> [\<psi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"
pdg_vsubset_index_pdigraph: fixes J :: "V" assumes "J \<subseteq>\<^sub>\<circ> J \<union>\<^sub>\<circ> K" shows "pdigraph \<alpha> J \<AA>"
fc_traces: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and u :: "'b"   and xs :: "'a list"   and ys :: "'a list" assumes "future_consistent P D R"   and "u \<in> range D"   and "(xs, ys) \<in> R u" shows "(xs \<in> traces P) = (ys \<in> traces P)"
in_above: fixes a :: "alt"   and b :: "alt"   and L :: "(alt \<times> alt) set"   and x :: "alt"   and y :: "alt" assumes "a \<noteq> b"   and "L \<in> Lin" shows "(x <\<^bsub>Arrow_Order.above L a b\<^esub> y) = (x \<noteq> y \<and> (if x = b then a <\<^bsub>L\<^esub> y else if y = b then x = a \<or> x <\<^bsub>L\<^esub> a else x <\<^bsub>L\<^esub> y))"
fconverse_fconverse: shows "(r\<^sub>2\<inverse>\<^sub>\<bullet>)\<inverse>\<^sub>\<bullet> = r\<^sub>2"
lookup_single_not_eq: fixes k :: "'d"   and k' :: "'d"   and v :: "'e" assumes "k \<noteq> k'" shows "lookup (monomial v k) k' = (0::'e)"
find_graph_occurence: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and E :: "('e \<times> 'g \<times> 'g) set"   and V :: "'g set"   and f :: "('g \<times> 'f) set" assumes "chain S"   and "finite E"   and "finite V"   and "graph_homomorphism (LG E V) (chain_sup S) f" shows "\<exists>i. graph_homomorphism (LG E V) (S i) f"
vector_inf_one_comp: fixes x :: "'a"   and y :: "'a" assumes "vector x" shows "(x \<sqinter> (1::'a)) * y = x \<sqinter> y"
assms: shows "\<eta>1 \<approx>>\<bullet> \<eta>2"
it_paths_RNodeE: fixes "is" :: "nat list"   and r :: "'a"   and ants :: "'a rose_tree list" assumes "is \<in> it_paths (RNode r ants)" obtains "isa__ = []"   | i :: "nat" and "is" :: "nat list" where "isa__ = i # is" and "i < length (children (RNode r ants))" and "is \<in> it_paths (children (RNode r ants) ! i)"
Fun_mkArr'_2: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "set_category_given_img.Fun S img (set_category_given_img.mkArr S img \<AA> A B F) = restrict F A"
REV_REV: fixes r :: "'c rexp" shows "REV (REV r) = r"
mojmir_to_rabin: fixes \<G> :: "'a ltl set"   and \<psi> :: "'a ltl" assumes "Only_G \<G>" shows "mojmir_to_rabin \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. \<G> \<up>\<Turnstile>\<^sub>P q}"
le_lbound_def: fixes c :: "'a"   and b :: "'a option" shows "c \<le>\<^sub>l\<^sub>b b = \<unlhd>\<^sub>l\<^sub>b (<) c b"
tb: shows "tm_bound t = round_interval prec (Ipoly (map2 (-) I (map interval_of xs)) (snd (tmf_polys z)))"
2: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "t \<in> set M"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "comp_timpl_closure {t} (set TI) = timpl_closure_set {t} (set TI)"
lookup_simps_3: fixes b :: "'b"   and es :: "'c \<Rightarrow> ('a, 'b, 'c) env option"   and x :: "'c"   and xs :: "'c list" shows "lookup (Env b es) (x # xs) = (case es x of None \<Rightarrow> None | Some e \<Rightarrow> lookup e xs)"
fls_deriv_mult_const_left: fixes c :: "'a"   and f :: "'a fls" shows "fls_deriv (fls_const c * f) = fls_const c * fls_deriv f"
insert: fixes A :: "'k set"   and x :: "'k" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
gsurjec_def: fixes F :: "('c, 'd) Group_scheme"   and G :: "('e, 'f) Group_scheme"   and f :: "'c \<Rightarrow> 'e" shows "gsurj\<^bsub>F,G\<^esub> f = (f \<in> gHom F G \<and> surj_to f (carrier F) (carrier G))"
ind_False: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<not> P n"   and "0 < n" shows "ind P n = (0::'a)"
unifier\<^sub>l\<^sub>s_def2: fixes L :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "L \<noteq> {}" shows "unifier\<^sub>l\<^sub>s \<sigma> L = (\<exists>l. L \<cdot>\<^sub>l\<^sub>s \<sigma> = {l})"
simps_5: fixes f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'e"   and x1 :: "'f" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
gauss_i_neq_1_1: shows "\<i>\<^sub>\<int> \<noteq> 1"
that: fixes I :: "real" assumes "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral I) {l..r}"   and "I \<le> 2 * c * (real n / real (n - 1)) / b ^ (n - 1)" shows "thesis"
alist_of_Abs_pq: fixes xs :: "('c \<times> 'd) list" assumes "distinct (map fst xs)"   and "sorted (map snd xs)" shows "alist_of (Abs_pq xs) = xs"
continuous_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" shows "continuous F f = (f \<longlongrightarrow> f (netlimit F)) F"
distinct_arcs_ex: fixes xs :: "nat list"   and i :: "nat"   and x :: "nat"   and j :: "nat" assumes "distinct xs"   and "i \<notin> set xs"   and "xs \<noteq> []" shows "\<exists>a b. a \<noteq> x \<and> (a, b) \<in> set (arcs i j xs)"
aw: shows "mus_adjuster f n_gs [] (0\<^sub>v n) = adjuster_wit [] f gs'"
wellformed_delta: fixes s :: "'s"   and a :: "'a" assumes "wellformed s"   and "a \<in> set alphabet" shows "wellformed (delta a s)"
level_bounded: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and k :: "nat" assumes "finite (nodes A)"   and "w \<notin> language A" obtains n :: "nat" where "\<And>l. n \<le> l \<Longrightarrow> card (level A w (2 * k) l) \<le> card (nodes A) - k"
bisims1'_eq_bisims1: fixes P :: "'a prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and n :: "nat" shows "bisims1'_syntax P es n h = bisims1_syntax P es h"
IH: fixes s :: "(latin_type \<Rightarrow> nat) set" assumes "latin_rect s (n - m) n"   and "m \<le> n" shows "\<exists>s'. s \<subseteq> s' \<and> latin_square s' n"
normalize_idem: fixes a :: "'a" shows "normalize (normalize a) = normalize a"
sep_add_left_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj a b"   and "sep_disj b c"   and "sep_disj a c" shows "plus b (plus a c) = plus a (plus b c)"
test_bit_bl: fixes w :: "'b word"   and n :: "nat" shows "w !! n = (rev (to_bl w) ! n \<and> n < size w)"
fp_cop_F_all: fixes d :: "'b"   and ds :: "'b set" assumes "d \<in> ds"   and "d \<notin> Xd ` cop ds" shows "Field (Pd d) \<subseteq> fp_cop_F ds"
a_ac_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y = y \<oplus> x"
polyhedron_Int_affine_minimal: fixes S :: "'b set" shows "polyhedron S = (\<exists>F. finite F \<and> S = affine hull S \<inter> \<Inter> F \<and> (\<forall>h\<in>F. \<exists>a b. a \<noteq> (0::'b) \<and> h = {x. a \<bullet> x \<le> b}) \<and> (\<forall>F'\<subset>F. S \<subset> affine hull S \<inter> \<Inter> F'))"
emeasure_Diff_null_set: fixes B :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "emeasure M (A - B) = emeasure M A"
non_comp_parallel: fixes u :: "'b list"   and v :: "'b list" shows "(\<not> u \<bowtie> v) = (u \<parallel> v)"