hcju/isabelleps huggingface dataset:
intro: fixes un :: "'c list \<Rightarrow> 'd \<Rightarrow> 'd"   and set_of :: "'d \<Rightarrow> 'c set"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empty :: "'d"   and succ :: "'c list \<Rightarrow> 'c list"   and rel :: "('c \<times> 'c) set" assumes "set_access un set_of memb empty"   and "set_access_succ_axioms succ rel" shows "set_access_succ set_of memb empty un succ rel"
lower_iso: shows "isotone \<X>\<^bsub>G \<circ>\<^sub>g F\<^esub> \<Y>\<^bsub>G \<circ>\<^sub>g F\<^esub> \<pi>\<^sup>*\<^bsub>G \<circ>\<^sub>g F\<^esub>"
root_poly_Re_def: fixes p :: "int poly" shows "root_poly_Re p = cf_pos_poly (poly_mult_rat (inverse 2) (poly_add p p))"
multithreaded_init_fin: shows "multithreaded init_fin_final init_fin"
right_unique_pcr_spmf_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_unique A" shows "right_unique (spmf_to_gpv.pcr_spmf_gpv A)"
da_good_approxE': fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and C :: "qtname"   and L :: "lenv"   and T :: "tys"   and A :: "assigned" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"   and "wf_prog G" obtains "normal s1 \<Longrightarrow> nrm A \<subseteq> dom (locals (snd s1))" and "\<And>l. \<lbrakk>fst s1 = Some (Jump (Break l)); normal s0\<rbrakk> \<Longrightarrow> brk A l \<subseteq> dom (locals (snd s1))" and "\<lbrakk>fst s1 = Some (Jump Ret); normal s0\<rbrakk> \<Longrightarrow> Result \<in> dom (locals (snd s1))"
ipresIGFresh_def: fixes h :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'c, 'k, 'l) model_scheme"   and MOD' :: "('m, 'n, 'g, 'o, 'p, 'j, 'd, 'q, 'r) model_scheme" shows "ipresIGFresh h MOD MOD' \<equiv> \<forall>ys y s X. igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFresh MOD' ys y (h X)"
lappend_snocL1_conv_LCons2: fixes xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "lappend (lappend xs (LCons y LNil)) ys = lappend xs (LCons y ys)"
annotateWhile: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and gs :: "('f \<times> 'e set) list"   and b :: "'e set"   and I :: "'e set"   and V :: "('e \<times> 'e) set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P whileAnnoG gs b I V c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P while gs b c Q,A"
attach_trace_eq': fixes \<I> :: "('h, 'i) \<I>"   and res1 :: "('h, 'i) resource"   and res2 :: "('h, 'i) resource"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('j, 'k, 'h, 'i) converter" assumes "outs_\<I> \<I> \<turnstile>\<^sub>R res1 \<approx> res2"   and "\<I> \<turnstile>res res1 \<surd>"   and "\<I> \<turnstile>res res2 \<surd>"   and "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>" shows "outs_\<I> \<I>' \<turnstile>\<^sub>R conv \<rhd> res1 \<approx> conv \<rhd> res2"
ipurge_tr_aux_append: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and ys :: "'b list" shows "ipurge_tr_aux I D U (xs @ ys) = ipurge_tr_aux I D U xs @ ipurge_tr_aux I D (sinks_aux I D U xs) ys"
FGModHom: shows "FGModuleHom G (\<cdot>) V (\<cdot>) T"
map_def: fixes f :: "'a" shows "\<ll>.map f = (if arr f then \<l>[cod f] \<cdot> T (\<I>, f) else null)"
pointwise_max_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "pointwise_max f g = (\<lambda>x. max (f x) (g x))"
cindex_poly_congr: fixes a :: "real"   and a' :: "real"   and b' :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < a'"   and "a' < b'"   and "b' < b"   and "\<forall>x. a < x \<and> x \<le> a' \<or> b' \<le> x \<and> x < b \<longrightarrow> poly p x \<noteq> 0" shows "cindex_poly a b q p = cindex_poly a' b' q p"
f: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "f \<in> Y" shows "current \<Gamma> f"
assms_2: shows "priK ` Ag \<inter> synth (analz H) = {}"
sum_lessThan_telescope': fixes f :: "nat \<Rightarrow> 'b"   and m :: "nat" shows "(\<Sum>n<m. f n - f (Suc n)) = f 0 - f m"
order_pderiv: fixes p :: "'b poly"   and a :: "'b" assumes "pderiv p \<noteq> 0"   and "order a p \<noteq> 0" shows "order a p = Suc (order a (pderiv p))"
conjunctive_monotonic: fixes S :: "'d \<Rightarrow> 'e" assumes "sconjunctive S" shows "mono S"
eq: shows "local.add a p\<^sub>1 p\<^sub>2 = local.add a p\<^sub>1 p\<^sub>3"
DBPass_def: shows "DBPass \<equiv> Deq inp q1 mid \<and> q2$ = $q2 @ [mid$] \<and> out$ = $out"
that: fixes y :: "'a" assumes "(x, y) \<in> A"   and "(y, z) \<in> A\<^sup>*" shows "thesis"
proper_it'D: fixes it :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and it' :: "'a \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'a" assumes "proper_it' it it'" shows "proper_it (it s) (it' s)"
p2: shows "prefix_of (psnoc \<pi> tdb) \<sigma>"
connected_component_eq: fixes y :: "'a"   and S :: "'a set"   and x :: "'a" assumes "y \<in> connected_component_set S x" shows "connected_component_set S y = connected_component_set S x"
runit_simps_5: fixes f :: "'d" assumes "D.ide f" shows "D.cod \<r>\<^sub>D[f] = f"
inter_scheme: fixes l :: "nat"   and U :: "nat list set" assumes "Form l U"   and "0 < l" obtains ka :: "nat" and kb :: "nat" and xs :: "nat list" and ys :: "nat list" where "l = ka + kb - 1" and "U = {xs, ys}" and "Form_Body ka kb xs ys (inter_scheme l U)" and "0 < kb" and "kb \<le> ka" and "ka \<le> Suc kb"
lessI: fixes n :: "nat" shows "n \<sqsubset> Suc n"
downset_faithful: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less"   and "ord.downset less_eq x \<subseteq> ord.downset less_eq y" shows "less_eq x y"
dgrad_p_set_le_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and F :: "('c \<Rightarrow>\<^sub>0 'f) set"   and G :: "('c \<Rightarrow>\<^sub>0 'f) set" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "gd_term.dgrad_p_set_le pair_of_term d F G = dgrad_set_le d (term_powerprod.pp_of_term pair_of_term ` Keys F) (term_powerprod.pp_of_term pair_of_term ` Keys G)"
rstar_below_sstar: fixes y :: "'a" assumes "has_least_prefixpoint (Rf y)"   and "has_least_prefixpoint (Sf y)" shows "rstar y \<le> sstar y"
select_convs_3: fixes 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 rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "rt \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = rt"
SUP_LE_P_def: fixes pair :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and X :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and u :: "'d"   and i :: "'c" shows "DiagramTermination.SUP_LE_P pair X u i = (SUP v\<in>{v. pair v i \<sqsubseteq> u}. X v i)"
approx_hIm: fixes x :: "complex star"   and y :: "complex star" assumes "x \<approx> y" shows "hIm x \<approx> hIm y"
cyclic_on_edge_succ: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d"   and x :: "'c" assumes "bidel_arc G M a"   and "x \<in> verts (bidel_arc.H G M a)"   and "out_arcs (bidel_arc.H G M a) x \<noteq> {}" shows "cyclic_on (edge_succ (bidel_arc.HM G M a)) (out_arcs (bidel_arc.H G M a) x)"
simps: fixes a :: "'a list" shows "walk a = (a = [] \<or> (\<exists>v. a = [v] \<and> v \<in> V) \<or> (\<exists>v w vs. a = v # w # vs \<and> v \<rightarrow> w \<and> walk (w # vs)))"
p_A'_def: fixes n :: "nat" shows "p_A' n = int (length (paid_A' ! n))"
a0n_def: shows "a0n \<equiv> \<lparr>init = a0n_init, trans = a0n_trans, obs = id\<rparr>"
m_lcomm: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier M"   and "y \<in> carrier M"   and "z \<in> carrier M" shows "x \<oplus>\<^bsub>M\<^esub> (y \<oplus>\<^bsub>M\<^esub> z) = y \<oplus>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> z)"
Ixy: shows "M \<le> Gromov_product_at px y y'"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "In1 x1 \<noteq> In2 x2"
simps: fixes gpv :: "('a, 'out, 'ret) gpv" shows "local.expectation_gpv gpv = \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. local.expectation_gpv (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None)"
lexordp_simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "ord.lexordp less xs [] = False"
sup_inf_distrib_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y \<cdot> z = (x + y) \<cdot> (x + z)"
signed_take_bit_int_less_eq_self_iff: fixes n :: "nat"   and k :: "int" shows "(signed_take_bit n k \<le> k) = (- (2 ^ n) \<le> k)"
left_hom_axioms: shows "left_hom (\<cdot>) (\<star>) a"
equality: fixes r :: "'a view_scheme"   and r' :: "'a view_scheme" assumes "ext r = ext r'"   and "lan r = lan r'"   and "own r = own r'"   and "more r = more r'" shows "r = r'"
list_emb_code_3: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "list_emb P (x # xs) (y # ys) = (if P x y then list_emb P xs ys else list_emb P (x # xs) ys)"
H: shows "ptrm_infer_type \<Gamma> (PApp A B) = Some \<tau>"
dgrad_p_setI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "\<And>v. v \<in> keys p \<Longrightarrow> d (pp_of_term v) \<le> m" shows "p \<in> dgrad_p_set d m"
rminusinf_uset: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and x :: "real" assumes "isrlfm p"   and "\<not> Ifm (a # bs) (minusinf p)"   and "Ifm (x # bs) p" shows "\<exists>(s, m)\<in>set (uset p). Inum (a # bs) s / real_of_int m \<le> x"
Q3: shows "Q3 \<in> carrier_mat (n - 1) (n - 1)"
mult_column_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and n :: "'e"   and q :: "'d" shows "mult_column A n q = (\<chi>i j. if j = n then A $h i $h j * q else A $h i $h j)"
f_i'_Suc: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "f_i' (Suc i) e = (if e \<in> \<^bold>E then f_i' i e + (k_i' i e - k_i' i (prod.swap e)) else 0)"
ell2_norm_geq0: fixes x :: "'c \<Rightarrow> complex" assumes "has_ell2_norm x" shows "0 \<le> ell2_norm x"
analytically_valid_imp_part_deriv_integrable_on: fixes s :: "(real \<times> real) set"   and f :: "real \<times> real \<Rightarrow> real"   and i :: "real \<times> real" assumes "analytically_valid s f i" shows "partial_vector_derivative f i integrable_on s"
P_\<phi>: fixes s :: "'s"   and t :: "'s" assumes "P s"   and "s \<in> \<phi>"   and "t \<in> set_pmf (K s)" shows "P t \<or> (AE \<omega> in T t. \<not> (HLD \<phi> suntil HLD \<psi>) (t ## \<omega>))"
map_of_clearjunk: fixes al :: "('a \<times> 'b) list" shows "map_of (AList.clearjunk al) = map_of al"
interval_bounds'_2: fixes a :: "'c"   and b :: "'c" assumes "cbox a b \<noteq> {}" shows "interval_lowerbound (cbox a b) = a"
lossless_geometric: fixes p :: "real" shows "lossless_spmf (geometric_spmf p) = (0 < p)"
pdevs_apply_pdevs_of_list: fixes xs :: "'b list"   and i :: "nat" shows "pdevs_apply (pdevs_of_list xs) i = (if i < length xs then xs ! i else (0::'b))"
Naa_def: shows "Naa = Neg ''a'' [Fun ''a'' []]"
diffs_of_real: fixes f :: "nat \<Rightarrow> real" shows "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
sorted_map_strict: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "strict_mono_on g {0..<n}" shows "sorted (map g [0..<n])"
mult_left_sub_dist_sup_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<le> x * (y \<squnion> z)"
assms_2: shows "theta \<subseteq> ZOretrT (theta \<union> ZObisT)"
Step1_4_4a1: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and l :: "Locs" shows "\<turnstile> $S4 rmhist p \<and> S4 rmhist p$ \<and> ReadInner rmCh mm ires p l \<and> unchanged (e p, c p, r p, rmhist!p) \<and> $MemInv mm l \<longrightarrow> ReadInner memCh mm (resbar rmhist) p l"
add_le_same_cancel2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a + b \<subseteq># b) = (a \<subseteq># {#})"
bt_dfs_Min_priority: fixes t :: "('c, 'd) bintree" assumes "is_heap_list (priority t) (children t)" shows "priority t = Min (set (bt_dfs priority t))"
correctCompositionSecrets_subcomp1: fixes C :: "specID"   and x :: "specID"   and s :: "Secrets" assumes "correctCompositionSecrets C"   and "x \<in> subcomponents C"   and "s \<in> specSecrets C" shows "\<exists>x\<in>subcomponents C. s \<in> specSecrets x"
fresh_def: fixes permute :: "perm \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "atom"   and x :: "'d" assumes "class.pt permute" shows "pt.fresh permute a x \<equiv> a \<notin> pt.supp permute x"
tt_plus_eqI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "tt p \<prec>\<^sub>t tt q" shows "tt (p + q) = tt p"
cqt_orig_2: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<psi> \<alpha>)) in v]"
isDERIV_Power: fixes x :: "nat"   and fa :: "floatarith"   and vs :: "real list"   and n :: "nat" assumes "isDERIV x fa vs" shows "isDERIV x (Power fa n) vs"
set_scale_spmf: fixes r :: "real"   and p :: "'c spmf" shows "set_spmf (scale_spmf r p) = (if 0 < r then set_spmf p else {})"
dma\<^sub>A_adaptation_eq: fixes mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and A\<^sub>C :: "'Var\<^sub>C \<Rightarrow> ('Val \<times> 'Val) option"   and x\<^sub>A :: "'Var\<^sub>A" shows "dma\<^sub>A mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>A_of A\<^sub>C] x\<^sub>A = dma\<^sub>C mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>C] (var\<^sub>C_of x\<^sub>A)"
surjective: fixes r :: "('b, 'c) tagged_packet_scheme" shows "r = \<lparr>p_iiface = p_iiface r, p_oiface = p_oiface r, p_src = p_src r, p_dst = p_dst r, p_proto = p_proto r, p_sport = p_sport r, p_dport = p_dport r, p_tcp_flags = p_tcp_flags r, p_payload = p_payload r, p_tag_ctstate = p_tag_ctstate r, \<dots> = tagged_packet.more r\<rparr>"
prems_2: shows "insert x F \<subseteq> carrier_vec n"
from_nat_to_nat_greatest: fixes k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec" shows "mod_type_class.from_nat (Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A))) = (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'d)"
Z: shows "Z = ((cmod z)\<^sup>2 - 1) / (1 + (cmod z)\<^sup>2)"
Un_Image: fixes R :: "('e \<times> 'd) set"   and S :: "('e \<times> 'd) set"   and A :: "'e set" shows "(R \<union> S) `` A = R `` A \<union> S `` A"
infsetsum'_converges_ennreal: fixes f :: "'c \<Rightarrow> ennreal"   and S :: "'c set" shows "infsetsum'_converges f S"
fps_mult_left_const_nth: fixes c :: "'a"   and f :: "'a fps"   and n :: "nat" shows "(fps_const c * f) $ n = c * f $ n"
cases: fixes x :: "'b hp option" obtains xa :: "'b" and uu_ :: "'b hp list" where "x = Some (Hp xa uu_)"   | "x = None"
phl_export2: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> w \<le> w \<cdot> z \<cdot> w"   and "\<lbrace>x\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>x\<rbrace>y \<cdot> w\<lbrace>z \<cdot> w\<rbrace>"
real_sqrt_inverse: fixes x :: "real" shows "sqrt (inverse x) = inverse (sqrt x)"
dom_eqI': fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "a \<in> domains f" shows "a = local.dom f"
m_closed: fixes M :: "('c, 'd) monoid_scheme"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Localization.submonoid M S"   and "x \<in> S"   and "y \<in> S" shows "x \<otimes>\<^bsub>M\<^esub> y \<in> S"
Arr_rep_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f" shows "monoidal_language.Arr C (free_monoidal_category.rep C f)"
lideal: fixes r :: "'b"   and a :: "'b" assumes "r \<in> B"   and "a \<in> f ` I" shows "multB r a \<in> f ` I"
Sup_least: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and 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 bot :: "'a"   and top :: "'a"   and A :: "'a set"   and z :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
Done: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" assumes "g s = Done" shows "productive_onp g s"
mult_double_iso: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "x \<le> y" shows "w \<cdot> x \<cdot> z \<le> w \<cdot> y \<cdot> z"
simps_2: fixes old :: "nat"   and v :: "nat"   and f :: "face"   and i :: "nat"   and "is" :: "nat list" shows "natToVertexListRec old v f (i # is) = (if i = old then None # natToVertexListRec i v f is else Some f\<^bsup>i\<^esup> \<bullet> v # natToVertexListRec i v f is)"
partition_on_mset_all: fixes A :: "'b multiset" assumes "A \<noteq> {#}" shows "partition_on_mset A {#A#}"
set_plus_elim: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "x \<in> A + B" obtains a :: "'a" and b :: "'a" where "x = a + b" and "a \<in> A" and "b \<in> B"
normal_poly_general_coeff_mult: fixes f :: "real poly"   and h :: "nat"   and j :: "nat" assumes "normal_poly f"   and "h \<le> j" shows "coeff f h * coeff f (j + 2) \<le> coeff f (h + 1) * coeff f (j + 1)"
firstPosAppend: fixes a :: "'a"   and l :: "'a list"   and l' :: "'a list" assumes "a \<in> set l" shows "firstPos a l = firstPos a (l @ l')"
less_top_Value_Top_code: fixes x :: "'b" shows "(top.Value x < top.Top) = True"
SdlHPPR1: shows "SdlHPPB d PP R1"
is_float_simps_4: fixes p :: "nat"   and X :: "real" shows "is_float (truncate_down p X)"
MapsToMorDomCod_2: fixes f :: "'b"   and g :: "'b" assumes "f \<approx>> g" shows "dom (f ;; g) = dom f"
g: fixes l :: "nat"   and v :: "nat"   and i :: "nat"   and j :: "nat" shows "eval g [l, v, i, j] \<down>= e_snoc v (if l = 0 then j else if s (e_snoc v 0) \<noteq> s v then 0 else if s (e_snoc v 1) \<noteq> s v then 1 else z)"
strict_trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "strict_suffix b a"   and "strict_suffix c b" shows "strict_suffix c a"
assms_3: shows "\<forall>V\<in>rv S (CFG_node (sourcenode a)). state_val s V = state_val s' V"
divide_out_unique_nat_1: fixes p :: "nat"   and m :: "nat"   and n :: "nat"   and k :: "nat" assumes "\<not> is_unit p"   and "p \<noteq> 0"   and "\<not> p dvd m"   and "n = p ^ k * m" shows "m = n div p ^ multiplicity p n"
Times_Plus_TimesLR_PlusR: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Times_Plus (TimesLR r (Plus s t)) = Plus (Times_Plus (TimesLR r s)) (Times_Plus (TimesLR r t))"
times_bool_def: fixes a :: "bool"   and b :: "bool" shows "a * b = (a \<and> b)"
jordan_block_pow_dim_2: fixes n :: "nat"   and a :: "'a"   and r :: "nat" shows "dim_col (jordan_block n a ^\<^sub>m r) = n"
l_get_disconnected_nodes_axioms: shows "l_get_disconnected_nodes DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs"
distinct_letter_in: fixes w :: "'b list"   and a :: "'b" assumes "\<not> w \<in> [a]*" obtains m :: "nat" and b :: "'b" and q :: "'b list" where "[a] \<^sup>@ m \<cdot> [b] \<cdot> q = w" and "b \<noteq> a"
invfun_inj: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B" shows "inj_on (f\<inverse>\<^bsub>B,A\<^esub>) B"
measure_induct2: fixes meas :: "'a \<Rightarrow> 'b \<Rightarrow> nat"   and S :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'b" assumes "\<And>x1 x2. (\<And>y1 y2. meas y1 y2 < meas x1 x2 \<Longrightarrow> S y1 y2) \<Longrightarrow> S x1 x2" shows "S x1 x2"
funpow_dist1_prop1: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "(f ^^ n) x = y"   and "0 < n" shows "(f ^^ funpow_dist1 f x y) x = y"
alllsts_def: shows "alllsts \<equiv> \<lambda>A. {x. alllstsp (\<lambda>x. x \<in> A) x}"
assms: shows "atom g \<sharp> (f1, k1, f2, k2)"
callee_invariant_on_def: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>" shows "callee_invariant_on callee I \<I> \<equiv> (\<forall>s x y s'. (y, s') \<in> set_spmf (callee s x) \<longrightarrow> I s \<longrightarrow> x \<in> outs_\<I> \<I> \<longrightarrow> I s') \<and> (\<forall>s. I s \<longrightarrow> \<I> \<turnstile>c callee s \<surd>)"
Suffixes_def: fixes A :: "'b list set" shows "Suffixes A = {w. \<exists>q. q @ w \<in> A}"
convex_onD_Icc'': fixes x :: "real"   and y :: "real"   and f :: "real \<Rightarrow> real"   and c :: "real" assumes "convex_on {x..y} f"   and "c \<in> {x..y}" shows "f c \<le> (f x - f y) / (y - x) * (y - c) + f y"
dist_nz: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open" shows "(x \<noteq> y) = (0 < dist x y)"
dummy_stack_extended: fixes S :: "stack_elem list"   and x :: "stack_elem"   and S' :: "stack_elem list" assumes "set S \<subseteq> range Dummy"   and "x \<notin> range Dummy" shows "(S \<lesssim> x # S') = (S \<lesssim> S')"
assms_3: fixes y :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "P (y # xs)" shows "P (x # y # xs)"
assms_1: shows "Complete_Partial_Order.chain (\<le>) A"
simps: fixes n :: "nat" shows "\<UU>.\<S> n = {q. \<G> \<Turnstile>\<^sub>P Rep q} \<union> {q. \<exists>j\<ge>the \<UU>.smallest_accepting_rank. \<UU>.state_rank q n = Some j}"
intros_19: fixes \<Gamma> :: "'b formula list"   and \<AA> :: "'b formula"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<AA> # \<Theta>" shows "\<^bold>\<not> \<AA> # \<Gamma> \<Rightarrow> \<Theta>"
inj_def: fixes f :: "'q \<Rightarrow> 'r" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"