hcju/isabelleps huggingface dataset:
part1_Int_part2: fixes V :: "'a set" assumes "infinite V" shows "part1 V \<inter> part2 V = {}"
unchanged_safeI: fixes p :: "PrIds"   and t :: "state"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and s :: "state" assumes "(e p t, c p t, r p t, m p t, (rmhist!p) t) \<noteq> (e p s, c p s, r p s, m p s, (rmhist!p) s) \<Longrightarrow> (s, t) \<Turnstile> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" shows "(s, t) \<Turnstile> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"
group_eq_pow_eq_mod: fixes a :: "nat"   and b :: "nat" assumes "\<^bold>g [^] a = \<^bold>g [^] b"   and "0 < order \<G>" shows "[a = b] (mod order \<G>)"
ideD_1: fixes a :: "'c" assumes "\<tau>.A.ide a" shows "\<tau>.A.arr a"
ipresIGWlsAll_def: fixes h :: "'l \<Rightarrow> 'm"   and hA :: "'n \<Rightarrow> 'o"   and MOD :: "('p, 'q, 'r, 's, 't, 'u, 'l, 'n, 'v) model_scheme"   and MOD' :: "('w, 'x, 'r, 's, 'y, 'z, 'm, 'o, 'aa) model_scheme" shows "ipresIGWlsAll h hA MOD MOD' \<equiv> ipresIGWls h MOD MOD' \<and> ipresIGWlsAbs hA MOD MOD'"
1: shows "\<guillemotleft>induced_arrow a' \<chi>' : a' \<rightarrow> a\<guillemotright> \<and> D.cones_map (induced_arrow a' \<chi>') \<chi> = \<chi>'"
Rep_dense_inject: fixes x :: "'c dense"   and y :: "'c dense" shows "(Rep_dense x = Rep_dense y) = (x = y)"
assms_1: shows "Domainp cr\<^sub>1 = (\<lambda>x. x \<in> \<UU>\<^sub>1)"
monoE: fixes f :: "'a list \<Rightarrow> 'b"   and x :: "'a list"   and y :: "'a list" assumes "border_order.mono f"   and "border x y" obtains "f x \<le> f y"
mor_final: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" shows "mor UNIV UNIV s1 s2 car_final1 car_final2 str_final1 str_final2 (Equiv_Relations.proj (lsbis1 carT1 carT2 strT1 strT2) \<circ> beh1 s1 s2) (Equiv_Relations.proj (lsbis2 carT1 carT2 strT1 strT2) \<circ> beh2 s1 s2)"
ConjunctiveI: fixes times_abc :: "'e \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'e"   and Inf_b :: "'c set \<Rightarrow> 'c"   and Inf_c :: "'d set \<Rightarrow> 'd" assumes "\<And>A. times_abc a (Inf_b A) = Inf_c (times_abc a ` A)" shows "a \<in> Conjunctive.Conjunctive Inf_b Inf_c times_abc"
cases: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and a1 :: "'e"   and a2 :: "('d, 'e, 'f) psi"   and a3 :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(a1, a2, a3) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" obtains P :: "('d, 'e, 'f) psi" and \<Psi> :: "'e" and Q :: "('d, 'e, 'f) psi" where "a1 = \<Psi>" and "a2 = P" and "a3 = Q" and "assertionAux.FrameStatEq SImp' (assertionAux.insertAssertion SCompose' (assertionAux.extractFrame SCompose' SBottom' P) \<Psi>) (assertionAux.insertAssertion SCompose' (assertionAux.extractFrame SCompose' SBottom' Q) \<Psi>)" and "env.simulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq') Q" and "\<forall>\<Psi>'. (SCompose' \<Psi> \<Psi>', P, Q) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" and "(\<Psi>, Q, P) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
nfa_startnode_is_node: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat" assumes "wf_nfa A n" shows "nfa_is_node A (nfa_startnode A)"
neg_diag_empty_spec: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" assumes "Beta_Regions' X v n not_in_X"   and "i \<le> n"   and "M i i < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
impl_refine_intro: fixes P\<^sub>1 :: "'a upred"   and Q\<^sub>1 :: "'a upred"   and Q\<^sub>2 :: "'a upred"   and P\<^sub>2 :: "'a upred" assumes "Q\<^sub>1 \<sqsubseteq> P\<^sub>1"   and "P\<^sub>2 \<sqsubseteq> (P\<^sub>1 \<and> Q\<^sub>2)" shows "(P\<^sub>1 \<Rightarrow> P\<^sub>2) \<sqsubseteq> (Q\<^sub>1 \<Rightarrow> Q\<^sub>2)"
conc_epsilon_2: fixes A :: "'a list set" shows "A @@ {[]} = A"
inf_rr_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "pseudo_hoop_algebra.inf_rr times imp_r a b = times a (imp_r a b)"
comp_cod_arr: fixes f :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.VV.arr f"   and "C.VV.cod f = b" shows "C.VV.comp b f = f"
delete_point_strong_block_in_iff: fixes bl :: "'a set"   and p :: "'a" assumes "bl \<in># \<B>" shows "(bl \<in># str_del_point_blocks p) = (p \<notin> bl)"
sets_restrict_space: fixes M :: "'a measure"   and \<Omega> :: "'a set" shows "sets (restrict_space M \<Omega>) = sets.restricted_space M \<Omega>"
orderE: fixes b :: "'t"   and a :: "'t" assumes "b \<preceq>\<^sub>t a" obtains "a = ord_term_lin.max a b"
initI': fixes e :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "e_length e = Suc n"   and "\<And>x. x < Suc n \<Longrightarrow> f x \<down>= e_nth e x" shows "f \<triangleright> n = e"
hyps: shows "card {ps. length ps = n \<and> adj_path v ps} = k ^ n"
converged_ipd_same_icd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and m :: "nat"   and m' :: "nat"   and k :: "nat"   and k' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "l < m"   and "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m'"   and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"   and "l icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> (Suc k) = \<pi>' (Suc k')"   and "\<pi>' m' = ipd (\<pi> k)"   and "\<forall>n\<in>{k'..<m'}. \<pi>' n \<noteq> ipd (\<pi> k)" shows "\<exists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
assms_3: shows "Dom f = {} \<longrightarrow> Cod f = {}"
abs_of_nonpos: fixes a :: "'a" assumes "a \<le> (0::'a)" shows "\<bar>a\<bar> = - a"
covering: shows "U \<subseteq> \<Union> (cover ` index)"
sel_2: fixes x1 :: "'ah"   and x2 :: "'ag" shows "snd (x1, x2) = x2"
silent_moves_vpa_path_1: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,f \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "ms = targetnodes rs"   and "valid_return_list rs m"   and "length rs = length cs" shows "m -as\<rightarrow>* m'"
exhaust: fixes y :: "'b eclause" obtains x1 :: "'b literal set" and x2 :: "'b trm set" where "y = Ecl x1 x2"
set_intros_3: fixes x12 :: "'c"   and x11 :: "'d option" shows "x12 \<in> set2_list_op (Insert x11 x12)"
ereal_less_real_iff: fixes x :: "real"   and y :: "ereal" shows "(x < real_of_ereal y) = ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
Max_wrt_nonempty: assumes "alts \<noteq> {}"   and "finite alts" shows "Max_wrt (Pareto R) \<noteq> {}"
equalityI: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
simps: fixes a :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "(a \<in> SINK f) = (\<exists>x. a = x \<and> d_OUT f x = 0)"
wp_under_wlp_def: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "wp_under_wlp prog \<equiv> \<forall>P. unitary P \<longrightarrow> wp prog P \<tturnstile> wlp prog P"
sums_zeta_of_nat_offset: fixes n :: "nat"   and r :: "nat" assumes "1 < n" shows "(\<lambda>x. complex_of_real (1 / real ((r + x + 1) ^ n))) sums (zeta (of_nat n) - complex_of_real (\<Sum>k = 1..r. 1 / real (k ^ n)))"
fg_run_def: 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" shows "fg_run A rho HOs SHOs coords \<equiv> fg_init_config A (rho 0) (coords 0) \<and> (\<forall>i. \<exists>p. fg_next_config A p (HOs (round (rho i) p) p) (SHOs (round (rho i) p) p) (coords (round (rho (Suc i)) p) p) (rho i) (rho (Suc i))) \<and> (\<forall>p r. \<exists>n. round (rho n) p = r)"
tT: shows "\<Gamma>\<^sub>2 \<turnstile> t : T"
numeral_Bit1: fixes n :: "num" shows "numeral (num.Bit1 n) = numeral n + numeral n + (1::'a)"
a_set_child_nodes_def: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "CD.a_set_child_nodes ptr children = invoke CD.a_set_child_nodes_tups ptr children"
integrable_mult_indicator: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "A \<in> sets M"   and "integrable M f" shows "integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
eq_swap_execute: fixes w\<^sub>1 :: "'action list"   and p :: "'state"   and w\<^sub>2 :: "'action list" assumes "path w\<^sub>1 p"   and "w\<^sub>1 =\<^sub>S w\<^sub>2" shows "target w\<^sub>1 p = target w\<^sub>2 p"
acc_subseteq_bacc: fixes r :: "('a \<times> 'a) set" assumes "finite r" shows "Wellfounded.acc r \<subseteq> \<Union> (range (bacc r))"
integral_cos_nx: fixes n :: "int" shows "integral {- pi..pi} (\<lambda>x. cos (x * real_of_int n)) = (if n = 0 then 2 * pi else 0)"
R_def: shows "R = (\<lambda>x. A x - ln x)"
borel_measurable_stake: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and w :: "bool stream"   and y :: "bool stream" assumes "f \<in> borel_measurable (nat_filtration n)"   and "stake n w = stake n y" shows "f w = f y"
sb_compose1: fixes xs :: "'b"   and y2 :: "'a"   and x :: "'a"   and zs :: "'b"   and y1 :: "'a"   and z :: "'a" shows "z @zs[y1 / x]_xs @zs[y2 / x]_xs = z @zs[y1 @xs[y2 / x]_xs / x]_xs"
col_space_eq_row_space_transpose: fixes A :: "(('a, 'c) vec, 'b) vec" shows "col_space A = row_space (Finite_Cartesian_Product.transpose A)"
bl_bin_bl_rtf: fixes n :: "nat"   and bl :: "bool list" shows "bin_to_bl n (bl_to_bin bl) = rev (takefill False n (rev bl))"
zip_eq_conv: fixes xs :: "'g list"   and ys :: "'h list"   and zs :: "('g \<times> 'h) list" assumes "length xs = length ys" shows "(zip xs ys = zs) = (map fst zs = xs \<and> map snd zs = ys)"
content_eq_zero_iff: fixes p :: "'a poly" shows "(Polynomial.content p = (0::'a)) = (p = 0)"
ring_distrib2: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(y \<plusminus>\<^bsub>R\<^esub> z) \<cdot>\<^sub>r\<^bsub>R\<^esub> x = y \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<plusminus>\<^bsub>R\<^esub> z \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
nat_mult_leD1: fixes a :: "nat"   and b :: "nat" assumes "real a * real b \<le> x"   and "0 < b" shows "real a \<le> x"
csov: shows "s O ov \<subseteq> b \<union> m \<union> ov"
well_base\<^sub>h_mono_hmset: fixes M :: "hmultiset"   and N :: "hmultiset" assumes "local.well_base\<^sub>h M"   and "hmsetmset N \<subseteq># hmsetmset M" shows "local.well_base\<^sub>h N"
exp_conj_top_left: fixes P :: "'c \<Rightarrow> real" assumes "sound P" shows "\<guillemotleft> \<lambda>_. True \<guillemotright> && P = P"
star_of_simps_11: fixes x :: "'a"   and y :: "'a" shows "(star_of x < star_of y) = (x < y)"
step_3_similar: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat (??.Jordan_Normal_Form_Existence.step_3 A) A"
of_bool_eq_iff: fixes one :: "'a"   and zero :: "'a"   and p :: "bool"   and q :: "bool" assumes "class.zero_neq_one one zero" shows "(zero_neq_one.of_bool one zero p = zero_neq_one.of_bool one zero q) = (p = q)"
rfn_tool17: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a = b" shows "a - c = b - c"
cases: fixes x :: "('a \<times> 'b) list \<times> 'c" obtains xs :: "('a \<times> 'b) list" and ko :: "'c" where "x = (xs, ko)"
a_rcos_self: fixes x :: "int" assumes "x \<in> UNIV" shows "x \<in> Idl\<^bsub>\<Z>\<^esub> {m} +>\<^bsub>\<Z>\<^esub> x"
reachable_adj_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<rightarrow>\<^sup>*\<^bsub>G\<^esub> b"   and "b \<rightarrow>\<^bsub>G\<^esub> c" shows "a \<rightarrow>\<^sup>*\<^bsub>G\<^esub> c"
l_sort_form_def: fixes X :: "'b list \<times> 'b list \<times> 'b list" shows "l_sort_form X \<equiv> fst X = []"
match_list_sound: fixes d :: "'d \<Rightarrow> ('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list"   and \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term" assumes "match_list d P = Some \<sigma>" shows "\<sigma> \<in> matchers (set P)"
failure_divergence_refine_def: fixes P :: "'a process"   and Q :: "'a process" shows "P \<sqsubseteq>\<^sub>F\<^sub>D Q \<equiv> P \<le> Q"
first_edge_first: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and w' :: "'a" assumes "v \<leadsto>xs\<leadsto> w"   and "(v, w') \<in> edges_of_walk xs" shows "first_edge_of_walk xs = (v, w')"
bounded_interior: fixes S :: "'a set" assumes "bounded S" shows "bounded (interior S)"
map_id: fixes t :: "'a \<times> 'b" shows "map_prod id id t = t"
Ln_conv_Arctan: fixes z :: "complex" assumes "z \<noteq> - 1" shows "Ln z = - 2 * \<i> * Arctan (moebius 1 (- 1) (- \<i>) (- \<i>) z)"
nn_integral_indicator: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M" shows "integral\<^sup>N M (indicator A) = emeasure M A"
bigo_elt_subset: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(g)" shows "O(f) \<subseteq> O(g)"
MInfty_neq_ereal_2: fixes r :: "real" shows "- \<infinity> \<noteq> ereal r"
less_eq_mat_def: fixes A :: "'a mat"   and B :: "'a mat" shows "(A \<le> B) = (dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> (\<forall>i<dim_row B. \<forall>j<dim_col B. A $$ (i, j) \<le> B $$ (i, j)))"
b: shows "\<Sqinter> {e::'a::refinement_lattice. (c::'a::refinement_lattice) \<sqsubseteq> e \<parallel> (j::'a::refinement_lattice)} \<sqsubseteq> \<Sqinter> {f::'a::refinement_lattice. c \<sqsubseteq> f \<parallel> (i::'a::refinement_lattice)}"
B: assumes "CSP2 (R (r \<turnstile> p)) (a, b\<lparr>ok := False\<rparr>) \<or> CSP2 (R (r \<turnstile> p)) (a, b\<lparr>ok := True\<rparr>)" shows "Q"
listrel_iff_zip: fixes xs :: "'b list"   and ys :: "'c list"   and r :: "('b \<times> 'c) set" shows "((xs, ys) \<in> listrel r) = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). (x, y) \<in> r))"
inv: shows "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"
coeff_bound: shows "fds_nth F \<in> O(\<lambda>x. of_nat x powr complex_of_real (\<sigma> - 1))"
T_quasi_preserves_null_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "(A \<in> null_sets M) = (T --` A \<in> null_sets M)"
f_rev_chain_in_rtrancl: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and R :: "('b \<times> 'b) set" assumes "m \<le> n"   and "\<forall>i\<in>{m..<n}. (f (Suc i), f i) \<in> R" shows "(f n, f m) \<in> R\<^sup>*"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_same a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> chksubs (>\<^sub>t) t s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)))"
in_sccs_verts_conv_reachable: fixes S :: "'a set" shows "(S \<in> H.sccs_verts) = (S \<noteq> {} \<and> (\<forall>ua\<in>S. \<forall>v\<in>S. ua \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> v) \<and> (\<forall>ua\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> ua \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> ua))"
hta_idx_f_pres: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_has_idx_f (hta_ensure_idx_f H)"
consistent_list_set: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "(nat \<times> nat) list" assumes "abs_ast_prob problem"   and "ast_problem.wf_partial_state problem s" shows "list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') s) s"
independent_AbGroups_pairwise_int0: fixes Gs :: "'a set list"   and G :: "'a set"   and G' :: "'a set" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "G \<in> set Gs"   and "G' \<in> set Gs"   and "G \<noteq> G'" shows "G \<inter> G' = 0"
lideal_axioms: shows "lideal I R (+) (\<cdot>) \<zero> \<one>"
downset_upset_dual: shows "(`) \<partial> \<circ> \<down> = \<up> \<circ> \<partial>"
pcr_blinfun_def: fixes T :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and Ta :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "pcr_blinfun T Ta \<equiv> rel_fun T Ta OO cr_blinfun"
assms: shows "wfp_on (restrict_to P A)\<^sup>+\<^sup>+ A"
dot_lmul_matrix: fixes x :: "(real, 'b) vec"   and A :: "((real, 'a) vec, 'b) vec"   and y :: "(real, 'a) vec" shows "(x v* A) \<bullet> y = x \<bullet> (A *v y)"
max_ext_def: shows "max_ext \<equiv> \<lambda>R. {(xa, x). max_extp (\<lambda>x xa. (x, xa) \<in> R) xa x}"
U_2: shows "poincare_map p1 x0 \<in> U"
arcs_symmetric: fixes G :: "'b pair_pre_digraph"   and a :: "'b"   and b :: "'b" assumes "pair_sym_digraph G"   and "(a, b) \<in> parcs G" shows "(b, a) \<in> parcs G"
sorted_list_of_mset_set: fixes A :: "'d set" shows "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
elims: fixes x :: "'v \<times> 'v \<Rightarrow> ennreal"   and xa :: "'v vertex \<times> 'v vertex"   and y :: "ennreal" assumes "split x xa = y" obtains f :: "'v \<times> 'v \<Rightarrow> ennreal" and x :: "'v" and x' :: "'v" and y :: "'v" where "xa__ = fa__" and "xa = (Vertex xb__, Edge x' yb__)" and "ya__ = (if x' = xb__ then fa__ (xb__, yb__) else 0)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and x :: "'v" and y' :: "'v" and y :: "'v" where "xa__ = fb__" and "xa = (Edge x y', Vertex y)" and "ya__ = (if y' = y then fb__ (x, y) else 0)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and vb :: "'v" and vc :: "'v" and v :: "'v" and vd :: "'v" where "xa__ = fc__" and "xa = (Edge vb__ vc, Edge v vd)" and "ya__ = 0"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and va :: "'v" and vb :: "'v" where "xa__ = f" and "xa = (Vertex va, Vertex vb)" and "ya__ = 0"
equiv_domain_permutation: fixes A :: "'b set"   and B :: "'c set" shows "equiv (A \<rightarrow>\<^sub>E B) (domain_permutation A B)"
B_pos: fixes i :: "'a" shows "0 \<le> B i"
on_graph_id: fixes B :: "('b, 'a) labeled_graph" shows "on_graph B id = Id_on (vertices B)"
IH: fixes k :: "nat" assumes "k < n" shows "of_int (dirichlet_inverse f i k) = dirichlet_inverse (\<lambda>n. of_int (f n)) (of_int i) k"
classes_above_sees_field: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and F :: "char list"   and b :: "staticb"   and t :: "ty"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P \<turnstile> C sees F,b:t in C'" shows "P' \<turnstile> C sees F,b:t in C'"
pid_gstep_succI: fixes pid :: "nat"   and gc :: "('c, 'ls, 'gs) pid_global_config"   and a :: "'a"   and c' :: "'c"   and ls' :: "'ls"   and gs' :: "'gs" assumes "pid_valid gc pid"   and "cstep (cmd_of_pid gc pid) a c'"   and "en (ls_of_pid gc pid, pid_global_config.state gc) a"   and "ex (ls_of_pid gc pid, pid_global_config.state gc) a = (ls', gs')" shows "\<lparr>pid_global_config.processes = (pid_global_config.processes gc)[pid := \<lparr>command = c', state = ls'\<rparr>], state = gs'\<rparr> \<in> pid_gstep_succ gc"
acyclic_subset: fixes s :: "('b \<times> 'b) set"   and r :: "('b \<times> 'b) set" assumes "acyclic s"   and "r \<subseteq> s" shows "acyclic r"
piecewise_continuous_on_induct: fixes a :: "'c"   and b :: "'c"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" assumes "piecewise_continuous_on a b I f"   and "\<And>a b f. continuous_on {a..b} f \<Longrightarrow> P a b {} f"   and "\<And>a i b I f1 f2 f. \<lbrakk>a \<le> i; i \<le> b; i \<notin> I; P a i I f1; P i b I f2; piecewise_continuous_on a i I f1; piecewise_continuous_on i b I f2; \<And>x. \<lbrakk>a \<le> x; x < i\<rbrakk> \<Longrightarrow> f1 x = f x; \<And>x. \<lbrakk>i < x; x \<le> b\<rbrakk> \<Longrightarrow> f2 x = f x; a < i \<Longrightarrow> (f \<longlongrightarrow> f1 i) (at_left i); i < b \<Longrightarrow> (f \<longlongrightarrow> f2 i) (at_right i)\<rbrakk> \<Longrightarrow> P a b (insert i I) f"   and "\<And>a b i I f. \<lbrakk>P a b I f; finite I; i \<notin> I\<rbrakk> \<Longrightarrow> P a b (insert i I) f" shows "P a b I f"
Un: fixes S :: "'a set" assumes "open S" shows "\<exists>U\<subseteq>\<B>. S = \<Union> U"
inverse_eq_iff_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(inverse a = inverse b) = (a = b)"
scale_left_commute: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and x :: "'b" assumes "Modules.module scale" shows "scale a (scale b x) = scale b (scale a x)"
rotate11: fixes A11 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and A7 :: "fmla"   and A8 :: "fmla"   and A9 :: "fmla"   and A10 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A11 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H)))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H)))))))))) \<turnstile> B"
radical_idealI_internalized: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and x :: "'c" assumes "class.countable TYPE('c)"   and "class.linorder less_eq less"   and "finite X"   and "F \<subseteq> P[X]"   and "f \<in> P[X]"   and "x \<notin> X"   and "\<V> (insert (1 - punit.monom_mult (1::'d) (monomial 1 x) f) F) = {}" shows "f \<in> \<surd>ideal F"
monoD: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<le> f y"
akiyama_tanigawa_row_def: fixes n :: "nat"   and l :: "nat"   and u :: "nat" shows "akiyama_tanigawa_row n l u = map (akiyama_tanigawa n) [l..<u]"
delete_def: fixes ks :: "'c list"   and t :: "('c, 'd) trie" shows "delete ks t = Trie2.trie.Trie (delete_trie ks (impl_of t))"