hcju/isabelleps huggingface dataset:
set_awalk_verts_append: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and q :: "'b list"   and w :: "'a" assumes "awalk u p v"   and "awalk v q w" shows "set (awalk_verts u (p @ q)) = set (awalk_verts u p) \<union> set (awalk_verts v q)"
norm_FW_valid_preservation: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "valid_dbm M" shows "valid_dbm (DBM_Normalization.norm (FW M n) (\<lambda>x. real ((k \<circ> v') x)) n)"
pi: shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> P x = PP"
ipresIGCons_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'c, 'e, 'h) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'i, 'opSym, 'g, 'd, 'f, 'j) model_scheme" shows "ipresIGCons h hA MOD MOD' \<equiv> ipresIGVar h MOD MOD' \<and> ipresIGAbs h hA MOD MOD' \<and> ipresIGOp h hA MOD MOD'"
elims_2: fixes x :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme"   and xa :: "'varSort \<times> 'sort"   and xb :: "('varSort, 'k) sAbs" assumes "sWlsAbs x xa xb" obtains SEM :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme" and xs :: "'varSort" and s :: "'sort" and xs' :: "'varSort" and sF :: "'k \<Rightarrow> 'k" where "x = SEM" and "xa = (xs, s)" and "xb = sAbs xs' sF" and "isInBar (xs, s) \<and> xs = xs' \<and> (\<forall>sX. if sWls SEM (asSort xs) sX then sWls SEM s (sF sX) else sF sX = sDummy SEM s)"
IH: assumes "list_all (\<lambda>x. x = None) (map typeof xs)" shows "traverse cast_Dyn xs = Some (norm_stack xs)"
threshold: shows "\<bar>valof a / valof b\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
lc_mult_scalar_monomial_right: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c" shows "lc (p \<odot> monomial c v) = punit.lc p * c"
hnr_If: fixes \<Gamma> :: "assn"   and \<Gamma>1 :: "assn"   and a :: "bool"   and a' :: "bool"   and b' :: "'c Heap"   and \<Gamma>2b :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and b :: "'d nres"   and c' :: "'c Heap"   and \<Gamma>2c :: "assn"   and c :: "'d nres"   and \<Gamma>' :: "assn" assumes "\<Gamma> \<Longrightarrow>\<^sub>t \<Gamma>1 * hn_val bool_rel a a'"   and "a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') b' \<Gamma>2b R b"   and "\<not> a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') c' \<Gamma>2c R c"   and "\<Gamma>2b \<or>\<^sub>A \<Gamma>2c \<Longrightarrow>\<^sub>t \<Gamma>'" shows "hn_refine \<Gamma> (if a' then b' else c') \<Gamma>' R (if a then b else c)"
insert_seq_nonempty: fixes start :: "'a option"   and xs :: "('a \<times> 'a option) list" assumes "insert_seq start xs" shows "xs \<noteq> []"
arr_cod: fixes f :: "'j" assumes "J.arr f" shows "J.arr (J.cod f)"
sup_finfun_apply: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "($) (sup f g) = sup (($) f) (($) g)"
prj_in_hom_2: shows "\<guillemotleft>\<mu>\<nu>.prj\<^sub>0 : \<mu>.leg0 \<down>\<down> \<nu>.leg1 \<rightarrow>\<^sub>C \<nu>.apex\<guillemotright>"
sigma_algebra_Pow: fixes sp :: "'c set" shows "sigma_algebra sp (Pow sp)"
Inv_TensorDiag: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Can t"   and "Can u"   and "Diag t"   and "Diag u" shows "Inv (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Inv t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Inv u"
elims_2: fixes x :: "'b formula" assumes "is_lit_plus x" obtains "x = \<bottom>"   | "x = \<^bold>\<not> \<bottom>"   | uu_ :: "'b" where "x = Atom uu_"   | uv_ :: "'b" where "x = \<^bold>\<not> (Atom uv_)"
voters_def: fixes vs :: "nat \<Rightarrow> process \<Rightarrow> val option" shows "voters vs = {uu_. \<exists>a r v. uu_ = a \<and> ((r, a), v) \<in> map_graph (\<lambda>(x, y). vs x y)}"
star_denest: fixes x :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set" shows "t_star (x \<union> y) = t_star (t_prod (t_star x) (t_star y))"
usedBy_lit: fixes x :: "'e \<Longrightarrow> 'f"   and v :: "'g" shows "x \<natural> \<guillemotleft>v\<guillemotright>"
run_tell_writer: fixes return :: "'c \<times> 'b list \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> ('c \<times> 'b list \<Rightarrow> 'a) \<Rightarrow> 'a"   and w :: "'b"   and m :: "('b, 'c, 'a) writerT" shows "run_writer (tell_writer return bind w m) = bind (run_writer m) (\<lambda>(a, ws). return (a, w # ws))"
timpl_closure_set_timpls_trancl_subset: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and c :: "('c set \<times> 'c set) set" shows "timpl_closure_set M (c\<^sup>+) \<subseteq> timpl_closure_set M c"
seqI: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and g :: "'d" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.arr C g"   and "partial_magma.dom C g = partial_magma.cod C f" shows "partial_magma.arr C (C g f)"
extf_compat_list: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   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 y :: "('b, 'a) tm"   and x :: "('b, 'a) tm"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and f :: "'b"   and xs :: "('b, 'a) tm list"   and xs' :: "('b, 'a) tm list" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf"   and "y \<noteq> x"   and "gt y x" shows "extf f gt (xs @ y # xs') (xs @ x # xs')"
assms_4: shows "(x, y) = split_n_m_x (dim_row A\<^sup>T) (dim_col A\<^sup>T) X"
gen_gbg_impl_rel_eext_def: fixes Rm :: "('i \<times> 'k) set"   and Rf :: "('h \<times> 'j set set) set" shows "\<langle>Rm, Rf\<rangle>gen_gbg_impl_rel_eext = {(\<lparr>gbgi_F = Fi, \<dots> = mi\<rparr>, \<lparr>gbg_F = F, \<dots> = m\<rparr>) |Fi mi F m. (Fi, F) \<in> Rf \<and> (mi, m) \<in> Rm}"
intros_2: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" assumes "Formula.hereditarily_fs t\<^sub>\<alpha>" shows "Formula.hereditarily_fs (Formula.Not\<^sub>\<alpha> t\<^sub>\<alpha>)"
rotation_pi_2_x_axis: shows "moebius_circline (moebius_rotation (pi / 2)) x_axis = y_axis"
powsum_prod_suc: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "x\<^bsub>0\<^esub>\<^bsup>m\<^esup> \<cdot> x\<^bsub>0\<^esub>\<^bsup>Suc n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>Suc (m + n)\<^esup>"
Dag_update_rI: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag"   and q :: "ref"   and y :: "ref" assumes "Dag p l r t"   and "q \<notin> set_of t" shows "Dag p l (r(q := y)) t"
config'_append2: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)"   and s :: "'a \<times> 'b"   and xs :: "'c list"   and ys :: "'c list" shows "On_Off.config' step A s (xs @ ys) = On_Off.config' step A (On_Off.config' step A s xs) ys"
v_def: shows "v \<equiv> sdrop (Suc (length ys)) w"
condK_def: shows "condK \<equiv> \<Lambda> iK tK eK. bindK\<cdot>iK\<cdot>(\<Lambda> i. case i of ValKF\<cdot>f \<Rightarrow> \<bottom> | ValKTT \<Rightarrow> tK | ValKFF \<Rightarrow> eK | ValKN\<cdot>n \<Rightarrow> \<bottom>)"
order_refl: fixes a :: "real \<times> real"   and X :: "'a \<times> (real \<times> real) pdevs" assumes "a \<in> set (inl (snd X))" shows "ccw 0 a a"
qGood_qGoodInp: fixes delta :: "'c"   and qinp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and qbinp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs option" shows "qGood (qOp delta qinp qbinp) = (qGoodInp qinp \<and> qGoodBinp qbinp)"
axioms_3: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Set \<alpha> T" shows "vsv (T\<lparr>ArrVal\<rparr>)"
dominate_trans: fixes n1 :: "nat"   and n2 :: "nat"   and n3 :: "nat" assumes "dominate n1 n2"   and "dominate n2 n3" shows "dominate n1 n3"
mark_guards_simps_2: fixes f :: "'f"   and g :: "'d \<Rightarrow> 'd" shows "mark_guards f (Basic g) = Basic g"
share_def: fixes x :: "bool" shows "share x = coin_spmf \<bind> (\<lambda>a\<^sub>1. let b\<^sub>1 = x \<oplus> a\<^sub>1 in return_spmf (a\<^sub>1, b\<^sub>1))"
orthogonal_complement_def: fixes W :: "'a set" shows "orthogonal_complement W = {x. \<forall>y\<in>W. orthogonal x y}"
obs_postdominate: fixes n :: "'node"   and n' :: "'node"   and S :: "'node set" assumes "n \<in> obs n' (PDG_BS S)"   and "n \<noteq> (_Exit_)" shows "n postdominates n'"
cut_less_Min_eq: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>< t \<noteq> {}" shows "iMin (I \<down>< t) = iMin I"
assms_6: shows "hd (tl xs) = hd (tl P)"
birkhoff_aux1_1: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "{x \<in> space M. limsup (\<lambda>n. ereal (birkhoff_sum f n x)) = \<infinity>} \<in> sets Invariants"
s'_def: shows "s' \<equiv> \<lambda>i z. indicat_real (space M) z *\<^sub>R s i z"
e2_def: shows "e2 \<equiv> e1 / 2"
assms_1: shows "sf \<phi> \<subseteq> insert \<phi> X"
acc_induct_rule: fixes a :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<And>y. (y, x) \<in> r \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
Err_sup_Err: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'d err" shows "Err \<squnion>\<^bsub>Err.sup f\<^esub> x = Err"
Sup_is_action: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "(\<lambda>A. A \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` A}) \<in> {p. is_CSP_process p}"
bij: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "bij_betw bij_map \<V> \<V>'"
bre'_while_algo: fixes \<delta> :: "('d, 'e) ta_rule set"   and Qi :: "'d set" assumes "finite \<delta>" shows "while_algo (bre'_algo Qi \<delta>)"
If_def: fixes p :: "'b"   and c1 :: "'b"   and c2 :: "'b" shows "If p Then c1 Else c2 \<equiv> p \<cdot> c1 + ! p \<cdot> c2"
lex_scale1_zero: fixes v :: "real"   and u :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex (v *\<^sub>R u) 0 = (if 0 < v then Counterclockwise_2D_Arbitrary.lex u 0 else if v < 0 then Counterclockwise_2D_Arbitrary.lex 0 u else True)"
verit_eq_simplify_23: fixes a :: "'a" shows "(a = - a) = (a = (0::'a))"
differentiable_on_eq_differentiable_at: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "open s" shows "(f differentiable_on s) = (\<forall>x\<in>s. f differentiable at x)"
convex_lincomb_empty: fixes c :: "'a vec \<Rightarrow> 'a"   and x :: "'a vec" shows "\<not> convex_lincomb c {} x"
inducts: fixes P :: "enat \<Rightarrow> bool"   and enat :: "enat" assumes "\<And>nat. P (enat nat)"   and "P \<infinity>" shows "P enat"
is_empty_alt: fixes q :: "'a queue" shows "Optimized_MTL.is_empty q = (linearize q = [])"
numer_cf7_deriv_pos: fixes x :: "real" shows "0 \<le> numer_cf7_deriv x"
project_inject_cancel: fixes x :: "int" shows "project_Int_int (inject_Int_int x) = x"
IsZero_1: fixes \<rho> :: "nat \<rightarrow> ValD"   and \<rho>' :: "nat \<rightarrow> ('o ValK \<rightarrow> 'o) \<rightarrow> 'o" assumes "\<forall>v. \<eta>: \<rho>\<cdot>v \<mapsto> \<rho>'\<cdot>v" shows "\<eta>: \<lbrakk>e_\<rbrakk>\<rho> \<mapsto> evalK e_\<cdot>\<rho>'"
merge1: shows "\<forall>(q, \<tau>)\<in>set ps. merges f ps xs ! q = \<tau> \<squnion>\<^bsub>f\<^esub> xs ! q"
inf_sup_absorb: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "inf x (sup x y) = x"
lincomb_del2: fixes S :: "rat vec set"   and v :: "rat vec"   and a :: "rat vec \<Rightarrow> rat" assumes "finite S"   and "S \<subseteq> Rn"   and "True"   and "v \<in> S" shows "gs.lincomb a S = a v \<cdot>\<^sub>v v + gs.lincomb a (S - {v})"
infinite_iff_countable_subset: fixes S :: "'q set" shows "infinite S = (\<exists>f. inj f \<and> range f \<subseteq> S)"
trancl_image_by_rtrancl: fixes E :: "('a \<times> 'a) set"   and Vi :: "'a set" shows "E\<^sup>+ `` Vi \<union> Vi = E\<^sup>* `` Vi"
power_commutes: fixes a :: "'b list"   and n :: "nat" shows "a \<^sup>@ n \<cdot> a = a \<cdot> a \<^sup>@ n"
Least_def: fixes P :: "'a \<Rightarrow> bool" shows "Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
zero_order_5: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
intro: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and I :: "'c set" assumes "comm_ring R addition multiplication zero unit"   and "ideal I R addition multiplication zero unit"   and "pr_ideal_axioms R I multiplication" shows "pr_ideal R I addition multiplication zero unit"
trm_simps_3: fixes f :: "'a" assumes "arr f" shows "cod \<t>[f] = \<one>"
assms_2: shows "\<^bold>\<forall> p \<in> S"
final_no_red: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and t :: "'b"   and x :: "'a"   and m :: "'c"   and ta :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and x' :: "'a"   and m' :: "'c" assumes "multithreaded final r"   and "r t (x, m) ta (x', m')"   and "final x" shows "False"
mpoly_base_conv_1: fixes x :: "'b" shows "(0::'b) = poly 0 x"
tiny_dg_Obj_in_Vset: shows "\<AA>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
hoare_comp: fixes p :: "'b Assertion"   and x :: "'b"   and y :: "'b"   and q :: "'b Assertion" shows "hoare p (x * y) q = (\<exists>r. hoare p x r \<and> hoare r y q)"
arr_eqI: fixes t :: "('a E.term, 'a) arr"   and t' :: "('a E.term, 'a) arr" assumes "arr t"   and "arr t'"   and "Dom t = Dom t'"   and "Cod t = Cod t'"   and "Map t = Map t'" shows "t = t'"
prv_subst_scnj: fixes F :: "'fmla set"   and t :: "'trm"   and x :: "'var" assumes "F \<subseteq> fmla"   and "finite F"   and "t \<in> trm"   and "x \<in> var" shows "prv (eqv (subst (scnj F) t x) (scnj ((\<lambda>\<phi>. subst \<phi> t x) ` F)))"
badE: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a" assumes "bad P f" obtains "\<And>i j. i < j \<Longrightarrow> \<not> P (f i) (f j)"
hom_connected_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H" shows "(H \<nu> \<mu> \<noteq> partial_magma.null V) = (H \<nu> (partial_magma.dom V \<mu>) \<noteq> partial_magma.null V)"
mkCone_cone: fixes a :: "'a"   and \<pi> :: "'b \<Rightarrow> 'a" assumes "D1.cone a \<pi>" shows "D1.mkCone \<pi> = \<pi>"
RBT_Impl_rbt_all_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and rbt :: "('c, 'd) rbt" shows "RBT_Impl_rbt_all P rbt = (\<forall>(k, v)\<in>set (RBT_Impl.entries rbt). P k v)"
tStep_is_tMap_conv: fixes tp :: "'a Trace"   and s :: "'a"   and f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace" shows "(tp \<leadsto> s = tMap f t) = (\<exists>tp' s'. t = tp' \<leadsto> s' \<and> s = f s' \<and> tp = tMap f tp')"
stream_space_borel_union: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and i :: "nat"   and n :: "nat"   and A :: "'a set" assumes "borel_adapt_stoch_proc F X"   and "i \<le> n"   and "A \<in> sets borel" shows "\<forall>y\<in>A \<inter> range (X i). X i -` {y} = proj_stoch_proc X n -` \<Union> (stream_space_single (proj_stoch_proc X n) ` comp_proj_i X n i y)"
elims: fixes x :: "nat"   and xa :: "atom list"   and xb :: "atom fm list"   and y :: "atom fm option" assumes "luckyFind x xa xb = y" obtains v :: "nat" and L :: "atom list" and F :: "atom fm list" where "x = v" and "xa = L" and "xb = F" and "y = (case find_lucky_eq v L of None \<Rightarrow> None | Some p \<Rightarrow> Some (elimVar v L F (Eq p)))"
uniform_groups: fixes G :: "'a set" assumes "G \<in> \<G>" shows "card G = \<m>"
ineq1: shows "cs q \<bullet> b < (0::'a)"
vec_of_basis_enum_ell2_component: fixes i :: "nat"   and \<psi> :: "'b ell2" assumes "i < CARD('b)" shows "vec_of_basis_enum \<psi> $ i = Rep_ell2 \<psi> (enum_class.enum ! i)"
Union_empty_conv: fixes A :: "'d set set" shows "(\<Union> A = {}) = (\<forall>x\<in>A. x = {})"
eqpoll_def: fixes A :: "'a set"   and B :: "'b set" shows "A \<approx> B \<equiv> \<exists>f. bij_betw f A B"
eff: fixes q :: "nat set"   and i :: "nat"   and j :: "nat" assumes "q \<in> P"   and "q \<noteq> I0"   and "i \<in> q"   and "j \<in> F q" shows "eff (fst cf1) (snd cf1) i \<approx> eff (fst cf2) (snd cf2) j"
Koszul_syz_sigs_auxI: fixes i :: "nat"   and j :: "nat"   and bs :: "('a \<Rightarrow>\<^sub>0 'b) list"   and k :: "nat" assumes "i < j"   and "j < length bs" shows "ord_term_lin.max (term_of_pair (punit.lt (bs ! i), k + j)) (term_of_pair (punit.lt (bs ! j), k + i)) \<in> set (Koszul_syz_sigs_aux bs k)"
spectrum_root_char_poly: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "spectrum A = {k. poly (char_poly A) k = (0::'b)}"
separating_RF_A: fixes G :: "('a, 'b) graph_scheme"   and A :: "'a set"   and B :: "'a set"   and X :: "'a set" assumes "separating_gen G A B X" shows "A \<subseteq> roofed_gen G B X"
indep_sets_mono_sets: fixes F :: "'b \<Rightarrow> 'a set set"   and I :: "'b set"   and G :: "'b \<Rightarrow> 'a set set" assumes "indep_sets F I"   and "\<And>i. i \<in> I \<Longrightarrow> G i \<subseteq> F i" shows "indep_sets G I"
first_entrance_meas: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "first_entrance_set A n \<in> sets M"
IH: fixes s1 :: "'s1"   and s2 :: "'s2"   and gpv1 :: "('a, 'out, 'in) gpv"   and gpv2 :: "('b, 'out, 'in) gpv" assumes "S s1 s2"   and "eq_\<I>_gpv A \<I> gpv1 gpv2" shows "rel_spmf (rel_sum (rel_prod A S) (\<lambda>a. case a of (q, a) \<Rightarrow> case a of (rpv1, rpv2) \<Rightarrow> \<lambda>a. case a of (q', rpv1', rpv2') \<Rightarrow> q = q' \<and> q' \<in> outs_\<I> \<I>' \<and> (\<exists>q''\<in>outs_\<I> \<I>. (\<forall>r\<in>responses_\<I> \<I>' q'. eq_\<I>_gpv (rel_prod (eq_onp (\<lambda>r'. r' \<in> responses_\<I> \<I> q'')) S) \<I>' (rpv1 r) (rpv1' r)) \<and> (\<forall>r'\<in>responses_\<I> \<I> q''. eq_\<I>_gpv A \<I> (rpv2 r') (rpv2' r'))))) (inline1' gpv1 s1) (inline1'' gpv2 s2)"
exists_extension_with_roots: shows "\<exists>L\<in>extensions. \<forall>P\<in>carrier (poly_ring R). ring.splitted L (\<sigma> P)"
disj_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(x OR y) OR z = x OR y OR z"
IH: fixes y :: "('b \<times> ('a \<times> 'b) list) \<times> 'b \<times> ('a \<times> 'b) list" assumes "(y, (s, [\<alpha>_step] @ \<rho>_step), s, [\<beta>_step] @ \<upsilon>_step) \<in> pex r"   and "peak ars y" shows "\<exists>\<sigma>' \<tau>'. DD ars r (fst y, snd y, \<sigma>', \<tau>')"
longest_common_prefix: fixes xs :: "'g list"   and ys :: "'g list" shows "\<exists>ps xs' ys'. xs = ps @ xs' \<and> ys = ps @ ys' \<and> (xs' = [] \<or> ys' = [] \<or> hd xs' \<noteq> hd ys')"
K_inconsistent_neg: fixes p :: "'b fm" assumes "{p} \<notin> mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>})" shows "\<turnstile> (\<^bold>\<not> p)"
not_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "not one = uminus (numeral.numeral one plus (num.Bit0 num.One))"
flatten_sum: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b"   and I :: "'c set" shows "flatten (sum f I) = (\<Sum>i\<in>I. flatten (f i))"
simps_12: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y2 :: "'c" shows "\<not> rel_node R SOURCE (Inner y2)"
repc_def: fixes I :: "interp"   and f :: "char"   and d :: "real" shows "repc I f d \<equiv> mkinterp (\<lambda>c. if c = f then d else Consts I c, Funcs I, Preds I, Games I)"
intro: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "2 < p"   and "prime q"   and "2 < q"   and "p \<noteq> q" shows "QR p q"
degree_diff_cancel: fixes q :: "'b poly"   and p :: "'b poly" assumes "degree q < degree p" shows "degree (p - q) = degree p"
measure_begin_induct: fixes P :: "nat \<times> cell list \<times> cell list \<Rightarrow> bool"   and f :: "nat \<Rightarrow> nat \<times> cell list \<times> cell list" assumes "\<And>n. \<not> P (f n) \<Longrightarrow> (f (Suc n), f n) \<in> measure_begin" shows "\<exists>n. P (f n)"
points_index_alt_def: fixes ps :: "('a \<times> 'b) set" shows "f bl index ps = size (filter_mset ((\<subseteq>) ps) (f bl))"
Cd_Xd: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Cd d X" shows "Xd x = d"
rot_img_eq: shows "cubeImage diamond_cube_gen = cubeImage rot_diamond_cube"
elims: fixes x :: "atom"   and y :: "int" assumes "divisor x = y" obtains i :: "int" and ks :: "int list" where "x = Le i__ ks__" and "y = 1"   | d :: "int" and i :: "int" and ks :: "int list" where "x = Dvd d__ ia__ ksa__" and "y = d__"   | d :: "int" and i :: "int" and ks :: "int list" where "x = NDvd d i ks" and "y = d"
polyn_deg_add1: fixes p :: "'a"   and q :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>"   and "q \<in> carrier R"   and "q \<noteq> \<zero>"   and "deg_n R S X p < deg_n R S X q" shows "deg_n R S X (p \<plusminus> q) = deg_n R S X q"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "comp x y = Lt"   and "comp y z = Lt" shows "comp x z = Lt"
fls_zero_eqI: fixes f :: "'a fls" assumes "\<And>n. fls_nth f n = (0::'a)" shows "f = 0"
arr_char: fixes f :: "'a" shows "Cop.arr f = A.arr f"
ESem_simps_1: fixes x :: "Vars.var"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> Lam [x]. e \<rbrakk>\<^bsub>\<rho>\<^esub> = Fn\<cdot>(\<Lambda> v. \<lbrakk> e \<rbrakk>\<^bsub>\<rho>(x := v)\<^esub>)"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "inf_class.inf a b \<le> b"
H2_J_2_a: fixes P :: "'c \<times> 'd alpha_d_scheme \<Rightarrow> bool"   and a :: "'c"   and b :: "'d alpha_d_scheme" shows "P (a, b) \<longrightarrow> (P ;; J) (a, b)"
AF_non_Nil: fixes k :: "nat"   and i :: "nat" shows "list_of (AF k i) \<noteq> []"
relpow_0_I: fixes x :: "'a"   and R :: "('a \<times> 'a) set" shows "(x, x) \<in> R ^^ 0"
ereal_divide_pos_iff: fixes x :: "ereal"   and y :: "ereal" shows "(0 < x / y) = ((y \<noteq> \<infinity> \<and> y \<noteq> - \<infinity>) \<and> (0 < x \<and> 0 < y \<or> x < 0 \<and> y < 0 \<or> y = 0 \<and> 0 < x))"