hcju/isabelleps huggingface dataset:
head: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "sum g {m..n} = g m + sum g {m<..n}"
"1_1": fixes n :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj n A) \<subseteq> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
split_sel_asm: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and option :: "'g option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> option = Some (the option) \<and> \<not> P (f2 (the option))))"
subst_apply_stateful_strand_def: fixes S :: "('c, 'd) stateful_strand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
basis_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> insert x X"   and "X_insert.basis_in \<E> X" shows "X \<subseteq> \<E>"
refl: fixes x :: "'a" shows "x \<sqsubseteq> x"
ordertype_inc_le_Ord: fixes A :: "'a set"   and \<alpha> :: "V"   and r :: "('a \<times> 'a) set"   and \<pi> :: "'a \<Rightarrow> V" assumes "small A"   and "Ord \<alpha>"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> r\<rbrakk> \<Longrightarrow> \<pi> x < \<pi> y"   and "wf r"   and "total_on A r"   and "\<pi> ` A \<subseteq> elts \<alpha>" shows "ordertype A r \<le> \<alpha>"
widen_Iface_Iface: fixes G :: "prog"   and I :: "qtname"   and J :: "qtname" assumes "G\<turnstile>Iface I\<preceq>Iface J" shows "G\<turnstile>I\<preceq>I J"
comp_in_eq_comp_out: shows "\<forall>x\<in>{0..1}. vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}) = vector_derivative \<gamma> (at x within {0..1}) \<bullet> i"
pinf_1: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and Q' :: "'a \<Rightarrow> bool" assumes "\<exists>z. \<forall>x>z. P x = P' x"   and "\<exists>z. \<forall>x>z. Q x = Q' x" shows "\<exists>z. \<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
trnr\<^sub>\<eta>_comp: fixes v :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "ide v"   and "seq \<mu> \<nu>"   and "src \<mu> = trg f" shows "trnr\<^sub>\<eta> v (\<mu> \<cdot> \<nu>) = (\<mu> \<star> f) \<cdot> trnr\<^sub>\<eta> v \<nu>"
lens_member_def: fixes x :: "'c \<Longrightarrow> 'd"   and A :: "'d scene" shows "(x \<in>\<^sub>S A) = ((\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on A \<triangleleft>\<^bsub>x\<^esub> s\<^sub>3 = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 \<triangleleft>\<^bsub>x\<^esub> s\<^sub>3 on A) \<and> (\<forall>b b'. get\<^bsub>x\<^esub> (b \<oplus>\<^sub>S b' on A) = get\<^bsub>x\<^esub> b'))"
poly_square_subst_def: fixes f :: "'a poly" shows "poly_square_subst f = poly_of_list (drop_half (coeffs f))"
simplex_cone: fixes p :: "nat"   and v :: "nat \<Rightarrow> real"   and l :: "nat \<Rightarrow> nat \<Rightarrow> real" shows "simplex_cone p v (oriented_simplex p l) = oriented_simplex (Suc p) (\<lambda>i. if i = 0 then v else l (i - 1))"
step_Normal_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Seq c1 c2, Normal s) \<rightarrow> u" obtains c\<^sub>1' :: "('e, 'd, 'f) com" and s' :: "('e, 'f) xstate" where "u = (Seq c\<^sub>1' c2, s')" and "\<Gamma>\<turnstile> (c1, Normal s) \<rightarrow> (c\<^sub>1', s')"   | "c1 = Skip" and "u = (c2, Normal s)"   | "u = (Throw, Normal s)" and "c1 = Throw"
succ: fixes as :: "'a list" shows "set (gen_succ rel as) = {b. \<exists>a\<in>set as. (a, b) \<in> set rel}"
n: shows "n \<in> Basis \<or> - n \<in> Basis"
eval_ztpoly_nonneg: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A" shows "0 \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p)"
PseudoHA_HAStates: fixes SA :: "('d, 'e, 'f) seqauto"   and D :: "'f data" shows "HAStates (PseudoHA SA D) = States SA"
assms_1: shows "\<forall>n. cls_val_process Mkt2 pf n \<in> borel_measurable (F n)"
resr_subdist_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<rightarrow> y \<le> x \<rightarrow> y \<squnion> z"
rep_eq: fixes x :: "int" shows "Rep_mod_ring (of_int_mod_ring x) = x mod int CARD('b)"
assms: fixes x :: "'a" assumes "x \<in> S" shows "(m1 f|` S) x \<sqsubseteq> (m2 f|` S) x"
hyps: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> length ys"   and "0 < n"   and "n' = length ys" shows "drop (n' - n) ys = rev ys ! (n - 1) # drop (Suc (n' - n)) ys"
rel_gpv'_eq: shows "spmf_to_gpv.rel_gpv' (=) = (=)"
check_addition_l_check_add: fixes A :: "(nat, (char list list \<times> int) list) fmap"   and B :: "(nat, int mpoly) fmap"   and r :: "(char list list \<times> int) list"   and r' :: "int mpoly"   and p :: "nat"   and p' :: "nat"   and q :: "nat"   and q' :: "nat"   and i :: "nat"   and i' :: "nat"   and \<V>' :: "char list set"   and \<V> :: "nat set"   and spec :: "(char list list \<times> int) list" assumes "(A, B) \<in> \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(r, r') \<in> sorted_poly_rel O mset_poly_rel"   and "(p, p') \<in> nat_rel"   and "(q, q') \<in> nat_rel"   and "(i, i') \<in> nat_rel"   and "(\<V>', \<V>) \<in> \<langle>var_rel\<rangle>set_rel" shows "check_addition_l spec A \<V>' p q i r \<le> \<Down> {(st, b). (\<not> is_cfailed st) = b \<and> (is_cfound st \<longrightarrow> spec = r)} (check_add B \<V> p' q' i' r')"
list_conj_qf: fixes ps :: "fm list" assumes "\<forall>p\<in>set ps. qfree p" shows "qfree (list_conj ps)"
minus_zero_vec: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v - 0\<^sub>v n = v"
f_a: shows "Id_on (vertices A) O (f_a \<union> f_b) = f_a"
integral_abs_bound: fixes M :: "'g measure"   and f :: "'g \<Rightarrow> real" shows "\<bar>integral\<^sup>L M f\<bar> \<le> LINT x|M. \<bar>f x\<bar>"
simps: fixes k :: "natural"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a"   and x :: "'b" shows "iterate k f x = (if k = 0 then Pair x else f x \<circ>\<rightarrow> iterate (k - 1) f)"
l_zero: fixes A :: "('h, 'i) aGroup_scheme"   and a :: "'h" assumes "aGroup A"   and "a \<in> carrier A" shows "\<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a"
HFail_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv4d s q" shows "HInv4d s' q"
Ext_ZF_def: fixes R :: "(ZF \<times> ZF) set"   and s :: "ZF" shows "Ext_ZF R s \<equiv> implode (Ext R s)"
of_weak_ranking_Cons: fixes z :: "'c set"   and zs :: "'c set list"   and y :: "'c"   and x :: "'c" shows "of_weak_ranking (z # zs) y x = (x \<in> z \<and> y \<in> \<Union> (set (z # zs)) \<or> of_weak_ranking zs y x)"
invariant_True: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "invariant rel (\<lambda>_. True)"
induct: fixes P :: "'c Tree \<Rightarrow> bool"   and a0 :: "'c Tree" assumes "\<And>v. P (T v E E)"   and "\<And>v va vb vc. \<lbrakk>P (T va vb vc); P (T va vb vc)\<rbrakk> \<Longrightarrow> P (T v (T va vb vc) E)"   and "\<And>v va vb vc. \<lbrakk>P (T va vb vc); P (T va vb vc)\<rbrakk> \<Longrightarrow> P (T v E (T va vb vc))"   and "\<And>v va vb vc vd ve vf. \<lbrakk>P (T va vb vc); P (T va vb vc)\<rbrakk> \<Longrightarrow> P (T v (T va vb vc) (T vd ve vf))"   and "\<And>v vd ve vf va vb vc. \<lbrakk>P (T vd ve vf); P (T vd ve vf)\<rbrakk> \<Longrightarrow> P (T v (T vd ve vf) (T va vb vc))"   and "P E" shows "P a0"
locate_in_limit: fixes x :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and X :: "'a set" assumes "x \<in> limit f X"   and "x \<notin> X" shows "\<exists>n. x \<in> funpower f (Suc n) X \<and> x \<notin> funpower f n X"
rep_eq: fixes x :: "('a, 'b) fps" shows "($) (x\<^sup>\<star>) = star_fps_rep (($) x)"
sign_def: fixes p :: "real Polynomial.poly"   and x :: "real" shows "sign p x \<equiv> if poly p x = 0 then 0 else if 0 < poly p x then 1 else - 1"
norm_Basis: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and Basis :: "'d set"   and u :: "'d" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "norm u = 1"
\<Gamma>_props_3: shows "tyenv_wellformed mds \<Gamma> \<S> P"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Arr (t \<^bold>\<cdot> u) = VSeq t u"
unique_connecting_path_properties_4: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "last (v \<leadsto> w) = w"
carrier_vs_is_self: fixes W :: "rat vec set" shows "carrier (gs.vs W) = W"
AE_pmfI: fixes M :: "'a pmf"   and P :: "'a \<Rightarrow> bool" assumes "\<And>y. y \<in> set_pmf M \<Longrightarrow> P y" shows "almost_everywhere (measure_pmf M) P"
comp: fixes Z :: "'d set set"   and g :: "'b \<Rightarrow> 'd" assumes "ChamberComplexMorphism Y Z g" shows "ChamberComplexMorphism X Z (g \<circ> f)"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and B :: "('a, 'b, 'c) boundOutput"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> ROut M B"   and "guarded P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) M B" shows "Prop C \<Psi> (!P) M B"
prime_power_not_one: fixes p :: "'a"   and k :: "nat" assumes "prime p"   and "0 < k" shows "p ^ k \<noteq> (1::'a)"
M_def: fixes \<pi> :: "(String.literal \<times> event_data list) prefix" shows "M \<pi> = {(i, v). i < pprogress \<phi> \<pi> \<and> wf_tuple (Formula.nfv \<phi>) (fv \<phi>) v \<and> (\<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> Formula.sat \<sigma> Map.empty (map the v) i \<phi>)}"
simps_10: fixes V :: "char list"   and T :: "ty"   and e :: "char list exp" shows "fv {V:T; e} = fv e - {V}"
card2_eq: fixes e :: "'a set" shows "(card e = 2) = (\<exists>u v. u \<noteq> v \<and> e = {u, v})"
in_keys_iff: fixes x :: "'d"   and s :: "'d \<Rightarrow>\<^sub>0 'e" shows "(x \<in> keys s) = (lookup s x \<noteq> (0::'e))"
var_dist_Nil: fixes t :: "dB" shows "var_dist [] t = t"
ideduct_synth_ik_replace: fixes M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "\<forall>t\<in>M. N \<turnstile>\<^sub>c t"   and "M \<turnstile>\<^sub>c t" shows "N \<turnstile>\<^sub>c t"
elims_3: fixes x :: "ivl"   and xa :: "ivl" assumes "\<not> le_aux x xa" obtains l1 :: "int option" and h1 :: "int option" and l2 :: "int option" and h2 :: "int option" where "x = I l1 h1" and "xa = I l2 h2" and "\<not> (le_option False l2 l1 \<and> le_option True h1 h2)"
simps_1: fixes guard :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf"   and s :: "'a" shows "loop_spmf.iter guard body 0 s = return_spmf s"
pprt_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.pprt zero sup x = sup x zero"
anno_type: fixes \<Gamma>' :: "Eg1.var \<Rightarrow> bexp set option"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and upd :: "Eg1.var ModeUpd"   and \<S>' :: "Eg1.var set \<times> Eg1.var set"   and P' :: "bexp set"   and P :: "bexp set"   and c :: "(Eg1.var, aexp, bexp) Stmt"   and \<Gamma>'' :: "Eg1.var \<Rightarrow> bexp set option"   and \<S>'' :: "Eg1.var set \<times> Eg1.var set"   and P'' :: "bexp set" assumes "\<Gamma>' = add_anno \<Gamma> \<S> upd"   and "\<S>' = add_anno_stable \<S> upd"   and "P' = restrict_preds_to_vars P {v. stable \<S>' v}"   and "has_type \<Gamma>' \<S>' P' c \<Gamma>'' \<S>'' P''"   and "c \<noteq> Stop"   and "\<And>x. subtype (to_total \<Gamma> x) P' (to_total \<Gamma>' x)"   and "anno_type_stable \<Gamma> \<S> upd"   and "anno_type_sec \<Gamma> \<S> P upd" shows "has_type \<Gamma> \<S> P (c@[upd]) \<Gamma>'' \<S>'' P''"
max_mapping_def: fixes P :: "'b \<Rightarrow> 'a option"   and P' :: "'b \<Rightarrow> 'a option"   and x :: "'b" shows "max_mapping P P' x = (case (P x, P' x) of (None, None) \<Rightarrow> None | (None, Some x) \<Rightarrow> Map.empty x | (Some x, None) \<Rightarrow> None | (Some x, Some y) \<Rightarrow> Some (max x y))"
simps_2: fixes n :: "nat" shows "ordinal_of_nat (Suc n) = oSuc (ordinal_of_nat n)"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "R f = R f'" shows "f = f'"
add_diff_cancel_right': fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and fi :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "cancel_comm_semigroup f fi" shows "fi (f a b) b = a"
card_Union_disjoint: fixes C :: "'b set set" assumes "disjoint C"   and "\<And>A. A \<in> C \<Longrightarrow> finite A" shows "card (\<Union> C) = sum card C"
kuhn_induction: fixes p :: "nat"   and n :: "nat"   and lab :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" assumes "0 < p"   and "\<forall>x j. j \<le> n \<longrightarrow> (\<forall>j. x j \<le> p) \<and> x j = 0 \<longrightarrow> lab x j = 0"   and "\<forall>x j. j \<le> n \<longrightarrow> (\<forall>j. x j \<le> p) \<and> x j = p \<longrightarrow> lab x j = 1"   and "odd (card {s. ksimplex p n s \<and> (reduced n \<circ> lab) ` s = {..n}})" shows "odd (card {s. ksimplex p (Suc n) s \<and> (reduced (Suc n) \<circ> lab) ` s = {..Suc n}})"
assms_1: shows "x \<in> space\<^sub>N N"
left_neutral: fixes a :: "'a" shows "(0::'a) OR a = a"
ltlc_aps_list'_correct: fixes \<phi> :: "'a ltlc"   and l :: "'a list" shows "set (ltlc_aps_list' \<phi> l) = atoms_ltlc \<phi> \<union> set l"
hs_dfs_det_while_algo: fixes post :: "'a \<Rightarrow> 'a dlist"   and \<Sigma>i :: "('a, unit) HashMap.hashmap" assumes "finite ((hs_R post)\<^sup>* `` hs.\<alpha> \<Sigma>i)" shows "det_while_algo (hs_dfs_dwa \<Sigma>i post)"
val_Zp_ultrametric: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)" shows "min (padic_integers.val_Zp p x) (padic_integers.val_Zp p y) \<le> padic_integers.val_Zp p (x \<oplus>\<^bsub>padic_int p\<^esub> y)"
asym: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c" assumes "b \<prec>\<^sub>p a"   and "a \<prec>\<^sub>p b" shows "False"
pivot_positions_form_helper_1: fixes a :: "nat"   and b :: "nat"   and z :: "'b"   and A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "(a, b) \<in> set (pivot_positions_main_gen z A nr nc i j)" shows "i \<le> a"
simps_35: fixes x2 :: "'c" shows "set3_sum3 (In2 x2) = {}"
bij_betw_in_out_arcs: fixes v :: "'a" shows "bij_betw (\<lambda>(u, v). (v, u)) (arc_to_ends G ` in_arcs G v) (arc_to_ends G ` out_arcs G v)"
drefines_def: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "drefines \<phi> G A B \<equiv> \<forall>P Q. unitary P \<and> unitary Q \<and> P \<tturnstile> wp A Q \<longrightarrow> \<guillemotleft> G \<guillemotright> && (P \<circ> \<phi>) \<tturnstile> wp B (Q \<circ> \<phi>)"
Float_0_eq_0: fixes e :: "int" shows "Float 0 e = 0"
block_complement_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection" shows "incidence_system.block_complement point_set b \<equiv> point_set - b"
ring_times_minusl: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c" assumes "Ring R"   and "a \<in> carrier R" shows "-\<^sub>a\<^bsub>R\<^esub> a = (-\<^sub>a\<^bsub>R\<^esub> 1\<^sub>r\<^bsub>R\<^esub>) \<cdot>\<^sub>r\<^bsub>R\<^esub> a"
jordan_nf_via_factored_charpoly_def: fixes A :: "'b mat"   and es :: "'b list" shows "jordan_nf_via_factored_charpoly A es = triangular_to_jnf_vector (schur_upper_triangular A es)"
card_pairs: shows "card {(i, x). i \<in> I \<and> x \<in> A i} = card I * k"
\<Delta>'''_countable_network: shows "countable_network \<Delta>'''.\<Delta>'''"
E'_V2: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and V :: "'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "Subgraph_Node_Defs.E' E V x y" shows "V y"
rt_strictly_fresher_irefl: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "\<not> rt \<sqsubset>\<^bsub>dip\<^esub> rt"
topological_basis: fixes "open" :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "class.topological_space open" shows "topological_space.topological_basis open B = (\<forall>x. open x = (\<exists>B'\<subseteq>B. \<Union> B' = x))"
UnE': fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "x \<in> A \<union> B" obtains "x \<in> A"   | "x \<notin> A" and "x \<in> B"
b_contents: fixes bl :: "'a set" assumes "bl \<in># filter_mset ((\<in>) x) \<B>" shows "x \<in> bl"
"5_2": shows "length (filter P2 l'') \<le> k1"
uprod_split: fixes P :: "'b \<Rightarrow> bool"   and f :: "('c, 'b) commute"   and x :: "'c uprod" shows "P (case_uprod f x) = (\<forall>a b. x = Upair a b \<longrightarrow> P (apply_commute f a b))"
rel_dom_def: shows "rel_dom = {q. q ` G \<subseteq> punit.dgrad_p_set d m' \<and> r = (\<Sum>g\<in>G. q g \<odot> g)}"
PEI_4: fixes n :: "nat" shows "(\<lambda>i. ValN\<cdot>n) \<in> unlr PE.delta"
tm_append_second_halt_eq: fixes l :: "cell list"   and r :: "cell list"   and B :: "(action \<times> nat) list"   and stp :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and off :: "nat"   and A :: "(action \<times> nat) list" assumes "steps0 (Suc 0, l, r) B stp = (0, l', r')"   and "tm_wf0 B"   and "off = length A div 2"   and "is_even (length A)" shows "steps0 (Suc off, l, r) (A @ shift B off) stp = (0, l', r')"
obs_intraE: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n' :: "'b"   and n :: "'b"   and S :: "'b set" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "n' \<in> CFG.obs_intra sourcenode targetnode kind valid_edge n S" obtains as :: "'a list" where "CFG.intra_path sourcenode targetnode kind valid_edge n as n'" and "\<forall>nx\<in>set (CFG.sourcenodes sourcenode as). nx \<notin> S" and "n' \<in> S"
f2: shows "\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2)"
assms_1: shows "x \<sqsubset>\<^sup>i y"
iMOD_Suc: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]" shows "(Suc x \<in> [ r, mod m ]) = (m = Suc 0)"
idem: fixes a :: "'b multiset" shows "a \<union># a = a"
conds_1: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Minimal_Solutions" shows "cond_A x y"
chain_imp_not_SN_on: fixes f :: "nat \<Rightarrow> 'a"   and r :: "('a \<times> 'a) set"   and i :: "nat" assumes "\<forall>i. (f i, f (Suc i)) \<in> r" shows "\<not> SN_on r {f i}"
cont_fx: shows "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"
rbt_lookup_keys: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "dom (ord.rbt_lookup less t) = set (RBT_Impl.keys t)"
VerduennungListeA: fixes \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<Theta>" shows "\<Gamma> @ \<Gamma> \<Rightarrow> \<Theta>"
thetaSeqZO_ZOretr: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaSeqZO tval aval indis \<subseteq> PL_Indis.ZOretr tval aval indis (PL_Indis.thetaSeqZO tval aval indis \<union> PL_Indis.ZObis tval aval indis)"
dim_eq_card: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B :: "'e set"   and V :: "'e set" assumes "vector_space scale"   and "module.span scale B = module.span scale V"   and "\<not> module.dependent scale B" shows "vector_space.dim scale V = card B"
continuous_on_open_avoid: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and a :: "'c" assumes "continuous_on s f"   and "open s"   and "x \<in> s"   and "f x \<noteq> a" shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
assms: shows "P \<parallel> !P \<Longrightarrow>\<alpha> \<prec> P'"
s'_s_cfg_on: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.cfg_on s'" shows "repcs s cfg \<in> MDP.cfg_on s"
points_subset: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset" assumes "sub_set_system \<U> \<A> \<V> \<B>" shows "\<U> \<subseteq> \<V>"
Rep_upsets_inject: fixes x :: "'b upsets"   and y :: "'b upsets" shows "(Rep_upsets x = Rep_upsets y) = (x = y)"
liftAll2_def: fixes phi :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and inp1 :: "'g \<Rightarrow> 'e option"   and inp2 :: "'g \<Rightarrow> 'f option" shows "liftAll2 phi inp1 inp2 \<equiv> \<forall>i v1 v2. inp1 i = Some v1 \<and> inp2 i = Some v2 \<longrightarrow> phi v1 v2"
dvd_refl: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a a"
eql_cong1: fixes t :: "trm"   and t' :: "trm"   and u :: "trm"   and u' :: "trm" shows "{} \<turnstile> t EQ t' AND u EQ u' IMP t EQ u IMP t' EQ u'"
two_comparisons_into_case_order_6: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and Q :: "'b"   and R :: "'b"   and P :: "'b" assumes "comparator comp" shows "(if lt_of_comp comp x y then Q else if lt_of_comp comp y x then R else P) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
mat_inverse_simps_1: fixes c :: "'d" assumes "c \<noteq> (0::'d)" shows "mat (inverse c) ** mat c = mat (1::'d)"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys x = {k. lookup x k \<noteq> (0::'c)}"