hcju/isabelleps huggingface dataset:
induct: fixes valid_edge :: "'f \<Rightarrow> bool"   and src :: "'f \<Rightarrow> 'g"   and trg :: "'f \<Rightarrow> 'g"   and knd :: "'f \<Rightarrow> 'h edge_kind"   and E :: "'g"   and X :: "'g"   and x :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node"   and P :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node \<Rightarrow> bool" assumes "lift_valid_edge valid_edge src trg knd E X x"   and "\<And>a e. \<lbrakk>valid_edge a; src a \<noteq> E \<or> trg a \<noteq> X; e = (Node (src a), knd a, Node (trg a))\<rbrakk> \<Longrightarrow> P e"   and "\<And>e. e = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, Node E) \<Longrightarrow> P e"   and "\<And>e. e = (Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit) \<Longrightarrow> P e"   and "\<And>e. e = (NewEntry, (\<lambda>s. False)\<^sub>\<surd>, NewExit) \<Longrightarrow> P e" shows "P x"
cltn2_compose_right_abs: fixes M :: "((real, 3) vec, 3) vec"   and A :: "cltn2" assumes "invertible M" shows "cltn2_compose A (cltn2_abs M) = cltn2_abs (cltn2_rep A ** M)"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_backto_standard_pos_2_B ires rs (l, r) = (\<exists>ln rn. l = Bk # Bk \<up> ln @ Oc # ires \<and> r = Bk # Oc \<up> Suc (Suc rs) @ Bk \<up> rn)"
nospoof_def: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and rs :: "'b common_primitive rule list" shows "nospoof TYPE('a) iface ipassmt rs = (\<forall>p. approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface\<rparr>) rs Undecided = Decision FinalAllow \<longrightarrow> p_src p \<in> ipcidr_union_set (set (the (ipassmt iface))))"
Int_absorb: fixes A :: "'e set" shows "A \<inter> A = A"
simps_3: fixes x31 :: "'b trm"   and x32 :: "'b trm"   and y31 :: "'b trm"   and y32 :: "'b trm" shows "(x31 \<cdot> x32 = y31 \<cdot> y32) = (x31 = y31 \<and> x32 = y32)"
i_Exec_Stream_Init_strictly_causal: fixes n :: "nat"   and input1 :: "nat \<Rightarrow> 'd"   and input2 :: "nat \<Rightarrow> 'd"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "input1 \<Down> n = input2 \<Down> n" shows "i_Exec_Comp_Stream_Init trans_fun input1 c \<Down> Suc n = i_Exec_Comp_Stream_Init trans_fun input2 c \<Down> Suc n"
linorder_axioms: shows "class.linorder less_eq_char less_char"
tendsto_iff: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and F :: "'d filter" assumes "class.metric_space dist uniformity open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
runit_in_hom: fixes a :: "'c" assumes "C.ide a" shows "\<guillemotleft>\<r>\<^sub>C[a] : a \<otimes>\<^sub>C \<I>\<^sub>C \<rightarrow>\<^sub>C a\<guillemotright>"
poly_map_poly_eval_poly: fixes p :: "int poly poly" shows "poly (map_poly Ring_Hom_Poly.of_int_poly p) = eval_poly Ring_Hom_Poly.of_int_poly p"
sqn_invalidate_not_in_dests: fixes dip :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<notin> dom dests" shows "sqn (invalidate rt dests) dip = sqn rt dip"
vsub_eqvt: fixes pi :: "(name \<times> name) list"   and t :: "llam"   and n :: "nat"   and s :: "llam" shows "pi \<bullet> vsub t n s = vsub (pi \<bullet> t) (pi \<bullet> n) (pi \<bullet> s)"
affine_independent_span_gt: fixes S :: "'a set" assumes "\<not> affine_dependent S"   and "DIM('a) < card S" shows "affine hull S = UNIV"
welltyped: shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
cong_eval_child_high: fixes l :: "bdt"   and r :: "bdt"   and v :: "nat" assumes "l \<sim> r" shows "r \<sim> Bdt_Node l v r"
is_prefix_hd: fixes l :: "'a"   and vs :: "'a list" shows "is_prefix [l] vs = (l = hd vs \<and> vs \<noteq> [])"
bsNF: fixes j :: "nat" shows "NF (Var j \<degree>\<degree> map f bs')"
distinct_17: fixes x21 :: "exp"   and x22 :: "Vars.var"   and x61 :: "exp"   and x62 :: "exp"   and x63 :: "exp" shows "App x21 x22 \<noteq> (x61 ? x62 : x63)"
degree_eq_Suc_max: fixes x :: "'a pdevs" shows "degree x = (if \<forall>i. pdevs_apply x i = (0::'a) then 0 else Suc (Max {i. pdevs_apply x i \<noteq> (0::'a)}))"
closed_sum_zero_right: fixes A :: "'a set" shows "A +\<^sub>M {0::'a} = closure A"
imp: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla" shows "imp \<phi>1 \<phi>2 \<in> fmla"
weak_refl: fixes x :: "'a"   and y :: "'a" assumes "x .= y"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq> y"
rtos_hom2: fixes r :: "nat"   and s :: "nat"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "l \<le> r * s - Suc 0" shows "rtos r s l \<le> r * s - Suc 0"
nb\<^sub>3: shows "\<forall>v. (\<Psi>\<^sub>G\<^sub>+) v \<noteq> None \<longrightarrow> the ((\<Psi>\<^sub>G\<^sub>+) v) \<in> set (the (range_of \<Psi> v))"
\<beta>\<^sub>f_def: shows "\<beta>\<^sub>f = \<a>[tab\<^sub>1 f, f\<^sub>0gh\<^sub>1.p\<^sub>1, TTfgh_TfTgh.chine] \<cdot> TTfgh_TfTgh.the_\<nu> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>1 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]"
drop_keep_last: fixes u :: "'a list"   and n :: "nat"   and a :: "'a list"   and X :: "'a" assumes "u \<noteq> []"   and "u = drop n (a @ [X])" shows "u = drop n a @ [X]"
distinct_lists_disconnected_nodes: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "local.CD.a_distinct_lists h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "distinct disc_nodes"
reverse_subst_pres_denot: fixes v' :: "val"   and e' :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v :: "exp"   and y :: "nat"   and e :: "exp" assumes "v' \<in> E e' \<rho>"   and "is_val v"   and "e' = subst y v e" shows "\<exists>\<rho>' v''. v'' \<in> E v [] \<and> v' \<in> E e \<rho>' \<and> \<rho>' \<approx> (y, v'') # \<rho>"
cong_rotate_to: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "xs \<cong> rotate_to xs x"
dvd_triv_right: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times b a)"
set_of_k_words_length_m_def: fixes k :: "nat"   and m :: "nat" shows "set_of_k_words_length_m k m = {xk. xk \<in> {w. length w = k \<and> set w \<subseteq> L}} \<inter> cw_len_concat -` {m}"
apex_def: shows "Dom.apex \<equiv> Maps.dom Dom.leg0"
f_shrink_Nil: fixes k :: "nat" shows "[] \<div> k = []"
dis_log_alt_reductions_def: fixes \<G> :: "'a cyclic_group" shows "dis_log_alt_reductions \<G> \<equiv> dis_log_alt \<G> \<and> Cyclic_Group.cyclic_group \<G>"
matrix_def: fixes f :: "('e, 'f) vec \<Rightarrow> ('e, 'g) vec" shows "matrix f = (\<chi>i j. f (axis j (1::'e)) $ i)"
update_convs_16: fixes sip' :: "nat \<Rightarrow> nat"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "sip_update sip' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip' sip, \<dots> = more\<rparr>"
distinct_1: fixes x21 :: "color"   and x22 :: "('a, 'b) rbt"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "Empty \<noteq> Branch x21 x22 x23 x24 x25"
simps: fixes x :: "nat"   and y :: "nat"   and u :: "nat"   and v :: "nat" shows "plus_int_raw (x, y) (u, v) = (x + u, y + v)"
Ch_range_disjoint: fixes h :: "'h"   and h' :: "'h"   and A :: "'x set" assumes "h \<noteq> h'" shows "Ch h A \<inter> Ch h' A = {}"
assms: shows "\<exists>u. u dvd (1::'a) \<and> a = u * b"
simps_2: fixes TI :: "('c set \<times> 'c set) list"   and f :: "('d, 'e, 'c) prot_fun"   and T :: "(('d, 'e, 'c) prot_fun, 'f) Term.term list"   and g :: "('d, 'e, 'c) prot_fun"   and S :: "(('d, 'e, 'c) prot_fun, 'f) Term.term list" shows "timpls_transformable_to' TI (Fun f T) (Fun g S) = ((f = g \<or> is_Abs f \<and> is_Abs g \<and> in_trancl TI (the_Abs f) (the_Abs g)) \<and> list_all2 (timpls_transformable_to' TI) T S)"
ratfps_shift_def: fixes n :: "nat"   and x :: "'b ratfps" shows "ratfps_shift n x = (let (a, b) = quot_of_ratfps (x - ratfps_of_poly (ratfps_cutoff n x)) in quot_to_ratfps (poly_shift n a, b))"
lprefix_lappend_same: fixes xs :: "'a llist"   and ys :: "'a llist"   and zs :: "'a llist" shows "lprefix (xs @\<^sub>l ys) (xs @\<^sub>l zs) = (lfinite xs \<longrightarrow> lprefix ys zs)"
minimum_in: fixes i :: "nat_int" shows "i \<noteq> bot \<longrightarrow> minimum i \<^bold>\<in> i"
Ys_a: shows "set_mset Ys \<subseteq> A"
abs_eq: shows "local.upt_generator = Generator (\<lambda>n. n < bound, \<lambda>n. (n, Suc n))"
mono: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "delta_mono x"   and "delta_gt \<delta> y z"   and "(0::'a) \<le> x" shows "delta_gt \<delta> (x * y) (x * z)"
cop_F_rejected_inv_def: fixes ds :: "'b set"   and fp :: "'a set" shows "cop_F_rejected_inv ds fp = (\<forall>x\<in>Rf CH fp. \<exists>fp'\<subseteq>fp. x \<in> fp' \<and> above (Pd (Xd x)) x \<subseteq> fp' \<and> Xd x \<notin> Xd ` CH fp')"
cut_RF_separating: fixes \<Delta> :: "('a, 'b) network_scheme"   and V :: "'a set" assumes "separating_network \<Delta> V"   and "sink \<Delta> \<notin> V" shows "cut \<Delta> (RF\<^sup>N\<^bsub>\<Delta>\<^esub> V)"
last_def: shows "last \<equiv> rec_list undefined (\<lambda>x xs. If (xs = []) x)"
derivation_ge_Suc_j: shows "derivation_ge (drop n D) (Suc j)"
lc_factor: shows "Polynomial.smult (of_int l ^ n) (\<Prod>\<alpha>'\<in>Roots'. [:- \<alpha>', 1:]) = Ring_Hom_Poly.of_int_poly (Polynomial.smult lc_factor (\<Prod>P))"
2: shows "Some r = d\<^sub>2 basic_identity\<^sub>E"
isos_compose: fixes f :: "'c \<times> 'c"   and f' :: "'c \<times> 'c" assumes "C.VV.iso f"   and "C.VV.iso f'"   and "C.VV.seq f' f" shows "C.VV.iso (C.VV.comp f' f)"
unfold2_dtor: shows "unfold2 dtor1 dtor2 = id"
ex_less_of_nat_mult: fixes x :: "'b"   and y :: "'b" assumes "(0::'b) < x" shows "\<exists>n. y < of_nat n * x"
graph_symmetric: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs G" shows "(v, u) \<in> parcs G"
igSubstIPresIGWlsSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIPresIGWlsSTR MOD \<equiv> \<forall>ys Y y s X. igWls MOD s (igSubst MOD ys Y y X) = (igWls MOD (asSort ys) Y \<and> igWls MOD s X)"
sup_state_conv: fixes P :: "'a prog"   and ST :: "ty list"   and LT :: "ty err list"   and ST' :: "ty list"   and LT' :: "ty err list" shows "P \<turnstile> (ST, LT) \<le>\<^sub>i (ST', LT') = (P \<turnstile> ST [\<le>] ST' \<and> P \<turnstile> LT [\<le>\<^sub>\<top>] LT')"
iswitness: fixes r :: "nat"   and n :: "nat"   and X :: "nat set" assumes "r < n" shows "witness X n ! r = (r \<in> X)"
dgrad_p_set_closed_spoly: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m" shows "spoly p q \<in> dgrad_p_set d m"
binop_wf_dfa: fixes f :: "bool \<Rightarrow> bool \<Rightarrow> bool" shows "wf_dfa (binop_dfa f A B) n"
psp_stable2: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set" assumes "F \<in> A \<longmapsto> A'"   and "F \<in> stable B" shows "F \<in> B \<inter> A \<longmapsto> B \<inter> A'"
GF_advice_a3: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "X = \<F> \<phi> w"   and "X = \<G>\<F> \<phi> w" shows "w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>"
Translations_E: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "right_translations_of_group G composition unit"   and "f \<in> right_translations_of_group.translation G composition ` G" obtains x :: "'c" where "x \<in> G" and "f = right_translations_of_group.translation G composition x"
distrib_right: fixes a :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" shows "a \<oplus> b \<otimes> c = (a \<otimes> c) \<oplus> (b \<otimes> c)"
prems_1: shows "model valuation (filter (\<lambda>c. \<not> clauseTautology c) (c' # F'))"
same_set_semiJoin: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and x :: "nat set \<times> 'b option list set"   and other :: "nat set \<times> 'b option list" assumes "getIJ getIJ" shows "fst (New_max.semiJoin x other) = fst x"
i_length: shows "i < length [0..<max CARD('b) CARD('c)]"
collapse: fixes prod :: "'a \<times> 'c" shows "(fst prod, snd prod) = prod"
image_subset_iff: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
less: assumes "x < z"   and "z < y" shows "z \<in> C"
indRelRTPO_impl_TRel_is_weak_barbed_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_barbed_simulation (indRelRTPO TRel) (STCalWB SWB TWB)" shows "weak_barbed_simulation (TRel\<^sup>+) TWB"
rfloor_def: fixes x :: "real" shows "Multiseries_Expansion.rfloor x = real_of_int \<lfloor>x\<rfloor>"
inner_zero_left: fixes x :: "'b" shows "inner (0::'b) x = (0::'a)"
arctan_lower_44_def: shows "arctan_lower_44 \<equiv> \<lambda>x. pi / 2 - (256 + 5943 * x\<^sup>2 + 19250 * x ^ 4 + 15015 * x ^ 6) / (35 * x * (35 + 315 * x\<^sup>2 + 693 * x ^ 4 + 429 * x ^ 6))"
set_eqI': fixes A :: "'c set"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B"   and "\<And>x. x \<in> B \<Longrightarrow> x \<in> A" shows "A = B"
fthe_felem_eq: fixes x :: "'a" shows "fthe_elem {|x|} = x"
map2_map_map: fixes h :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and f :: "'j \<Rightarrow> 'h"   and xs :: "'j list"   and g :: "'j \<Rightarrow> 'i" shows "map2 h (map f xs) (map g xs) = map (\<lambda>x. h (f x) (g x)) xs"
distinct_11: fixes x3 :: "'b" shows "One \<noteq> Atom x3"
yu: shows "paid_A ! n ! (length (paid_A ! n) - Suc m) # drop (Suc (length (paid_A ! n) - Suc m)) (paid_A ! n) = drop (length (paid_A ! n) - Suc m) (paid_A ! n)"
radical_unique: fixes r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and b :: "'a fps"   and a :: "'a fps" assumes "r (Suc k) (fps_nth b 0) ^ Suc k = fps_nth b 0"   and "r (Suc k) (fps_nth b 0) = fps_nth a 0"   and "fps_nth b 0 \<noteq> (0::'a)" shows "(a ^ Suc k = b) = (a = fps_radical r (Suc k) b)"
fv_append: fixes x :: "'b list"   and y :: "'b list" shows "fv (x @ y) = fv x \<union> fv y"
size_list_estimation': fixes x :: "'b"   and xs :: "'b list"   and y :: "nat"   and f :: "'b \<Rightarrow> nat" assumes "x \<in> set xs"   and "y \<le> f x" shows "y \<le> size_list f xs"
poly_pos_less: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x<a. 0 < poly p x) = (p \<noteq> 0 \<and> poly_neg_inf p = 1 \<and> count_roots_below p a = (if poly p a = 0 then 1 else 0))"
case_cong: fixes lname :: "lname"   and lname' :: "lname"   and f1 :: "ename \<Rightarrow> 'a"   and g1 :: "ename \<Rightarrow> 'a"   and f2 :: "'a"   and g2 :: "'a" assumes "lname = lname'"   and "\<And>x1. lname' = EName x1 \<Longrightarrow> f1 x1 = g1 x1"   and "lname' = This \<Longrightarrow> f2 = g2" shows "(case lname of EName x1 \<Rightarrow> f1 x1 | This \<Rightarrow> f2) = (case lname' of EName x1 \<Rightarrow> g1 x1 | This \<Rightarrow> g2)"
left_right_simps_3: fixes t :: "'a" assumes "is_app t" shows "app (left t) (right t) = t"
not_simple_match_ip: fixes d :: "'a word"   and aa :: "simple_action" assumes "\<not> simple_match_ip (src m) s" shows "(runFw s d c (SimpleRule m aa # rs) = Decision FinalAllow) = (runFw s d c rs = Decision FinalAllow)"
assms_2: shows "y \<in> carrier (DirProds G I)"
groupWIs1_def: fixes c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list" shows "groupWIs1 c rs = (let P = getParts rs; W = map getOneIp P; f = \<lambda>wi. (map (\<lambda>d. runFw (getOneIp wi) d c rs) W, map (\<lambda>s. runFw s (getOneIp wi) c rs) W) in map (map fst) (groupF snd (map (\<lambda>x. (x, f x)) P)))"
winkerr: shows "\<alpha> \<squnion> \<beta> = \<beta>"
assms_3: shows "Rel O Rel' \<subseteq> Rel''"
assign: fixes x :: "'Var"   and e :: "'AExp"   and mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simple_abv (x \<leftarrow> e, mem) (Stop, mem(x := ev\<^sub>A mem e))"
fixp_preserves_mono1: fixes F :: "('c \<Rightarrow> 'd \<Rightarrow> 'e llist) \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. mono_llist_lift (\<lambda>f. F f x)"   and "f \<equiv> llist_lift.fixp_fun F"   and "\<And>f. monotone ord llist.le_fun f \<Longrightarrow> monotone ord llist.le_fun (F f)" shows "monotone ord llist.le_fun f"
set_tag_name_get_disconnected_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
sep_add_disjD: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj (plus x y) z"   and "sep_disj x y" shows "sep_disj x z \<and> sep_disj y z"
mpoly_Const_mult: fixes a :: "'b"   and b :: "'b" shows "Const (a * b) = Const a * Const b"
prime_dvd_fact_iff_int: fixes p :: "int"   and n :: "nat" assumes "prime p" shows "(p dvd fact n) = (p \<le> int n)"
eWlsBinp_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym" shows "\<not> eWlsBinp MOD delta (\<lambda>i. Some ERR)"
OPT2_C2: fixes x :: "'a"   and y :: "'a"   and qs :: "'a list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) = real (length qs - 2) / 2"
simps: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree" shows "pre_adjust \<langle>l, (a, lv), r\<rangle> = (invar l \<and> invar r \<and> (lv = lvl l + 1 \<and> (lv = lvl r + 1 \<or> lv = lvl r + 2 \<or> lv = lvl r \<and> sngl r) \<or> lv = lvl l + 2 \<and> (lv = lvl r + 1 \<or> lv = lvl r \<and> sngl r)))"
value_le: shows "value_flow \<Delta> g \<le> value_flow \<Delta> f"
not_subseteq: fixes A :: "'c set"   and B :: "'c set" assumes "\<not> A \<subseteq> B" shows "\<exists>a\<in>A. a \<notin> B"
res_e: shows "ord_resolve (S_M S M) CAs DA AAs As \<sigma> E"
root_main'_lower: fixes x :: "int"   and n :: "int"   and y :: "int"   and b :: "bool" assumes "0 \<le> x"   and "0 \<le> n"   and "root_main' x n = (y, b)" shows "y ^ p \<le> n"
cases: fixes a :: "'b" assumes "algebraic_int a" obtains p :: "'b poly" and x :: "'b" where "a = x" and "lead_coeff p = (1::'b)" and "\<forall>i. coeff p i \<in> \<int>" and "poly p x = (0::'b)"
InSync_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat"   and a :: "'addr" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (insync\<^bsub>V\<^esub> (a) e, xs) (insync\<^bsub>V\<^esub> (a) e', xs')"
left_alt: fixes j :: "nat" shows "local.left_spec j = (if local.has_left j then Some (local.the_left j) else None)"
subst_lam_eq: fixes x :: "nat"   and v :: "exp"   and e :: "exp" shows "subst x v (ELam x e) = ELam x e"
usubstappf_geq: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" assumes "usubstappt \<sigma> U \<theta> \<noteq> undeft"   and "usubstappt \<sigma> U \<eta> \<noteq> undeft" shows "usubstappf \<sigma> U (Geq \<theta> \<eta>) = Afml (Geq (the (usubstappt \<sigma> U \<theta>)) (the (usubstappt \<sigma> U \<eta>)))"
predicate2D_obj: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "P \<le> Q" shows "P x y \<longrightarrow> Q x y"
monoseq_Suc: fixes X :: "nat \<Rightarrow> 'b" shows "monoseq X = ((\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n))"
while_option_stop2: fixes b :: "'d \<Rightarrow> bool"   and c :: "'d \<Rightarrow> 'd"   and s :: "'d"   and t :: "'d" assumes "while_option b c s = Some t" shows "\<exists>k. t = (c ^^ k) s \<and> \<not> b t"
11: shows "\<forall>i j. i < row_length M1 * row_length M2 * row_length M3 \<and> j < length M1 * length M2 * length M3 \<longrightarrow> i mod row_length (M2 \<otimes> M3) < row_length M2 * row_length M3 \<and> j mod length (M2 \<otimes> M3) < length M2 * length M3"
uexpr_eq_iff: fixes e :: "('a, 'b) uexpr"   and f :: "('a, 'b) uexpr" shows "(e = f) = (\<forall>b. \<lbrakk>e\<rbrakk>\<^sub>e b = \<lbrakk>f\<rbrakk>\<^sub>e b)"
can_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b"   and "\<F>C.ide c" shows "\<F>C.can \<a>[a, b, c]"
conservative_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "conservative M T \<equiv> qmpt M T \<and> conservative_axioms M T"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "C.ide a"   and "C.seq f a" shows "C.dom f = a"
a: shows "a = \<^bold>\<langle>un_Prim a\<^bold>\<rangle>"
spaceN_sum: fixes I :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and N :: "'c quasinorm" assumes "\<And>i. i \<in> I \<Longrightarrow> x i \<in> space\<^sub>N N" shows "sum x I \<in> space\<^sub>N N"
bounded_antilinear_def: fixes f :: "'d \<Rightarrow> 'e" shows "bounded_antilinear f \<equiv> antilinear f \<and> bounded_antilinear_axioms f"
alls2_def: fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "alls2 A B a b = concat (map (\<lambda>ys. map (\<lambda>xs. (xs, ys)) (alls A a)) (alls B b))"
map_closed: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> translation ` G" shows "restrict (inv_into G translation) (translation ` G) a \<in> G"
FactGroup_inj_on: shows "inj_on (\<lambda>X. the_elem (h ` X)) (carrier (G Mod kernel G H h))"
converse3p_converse3p: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "converse3p (converse3p r) = r"