hcju/isabelleps huggingface dataset:
translate_eoracle_def: fixes translator :: "'r \<Rightarrow> 't \<Rightarrow> ('q list \<times> 'r) spmf"   and eoracle :: "'r \<times> 's \<Rightarrow> 'u \<Rightarrow> (('p \<times> 't list) \<times> 'r \<times> 's) spmf"   and state :: "'r \<times> 's"   and inp :: "'u" shows "translate_eoracle translator eoracle state inp \<equiv> bind_spmf (eoracle state inp) (\<lambda>((out, e_in), s). let conc = \<lambda>(es, st) e. map_spmf (map_prod ((@) es) id) (translator st e) in bind_spmf (foldl_spmf conc (return_spmf ([], fst s)) e_in) (\<lambda>(e_out, s_event). return_spmf ((out, e_out), s_event, snd s)))"
subst_vsubst_trans: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "good Y"   and "fresh ys y1 X" shows "X #[y1 // y]_ys #[Y / y1]_ys = X #[Y / y]_ys"
fiber_relation_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" shows "fiber_relation \<alpha> S T \<equiv> map \<alpha> S T"
cancel_term_pp_pair_def: fixes a :: "'t \<times> 'a" shows "cancel_term_pp_pair a = (fst a \<ominus> gcs (pp_of_term (fst a)) (snd a), snd a - gcs (pp_of_term (fst a)) (snd a))"
fold_mset_single: fixes s :: "'m"   and x :: "'m" shows "fold_mset funionM s {#x#} = funionM x s"
Nil_is_map_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "([] = map f xs) = (xs = [])"
T1: fixes ds :: "'b set"   and X :: "'a set" shows "stable_on ds X = (\<exists>XD_XH. stable_pair_on ds XD_XH \<and> X = match XD_XH)"
llength_lzip: fixes xs :: "'c llist"   and ys :: "'d llist" shows "llength (lzip xs ys) = min (llength xs) (llength ys)"
Cs_def: shows "Cs = map (\<lambda>C. THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C) Ds"
map_poly_map_poly: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and p :: "'e poly" assumes "f (0::'d) = (0::'c)" shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
dioph_ineq_dfs_bij: fixes n :: "nat"   and ks :: "int list"   and l :: "int"   and i :: "int"   and k :: "nat" shows "(fst (dioph_ineq_dfs n ks l) ! int_encode i = Some k \<and> dioph_is_node ks l i) = (k < length (snd (dioph_ineq_dfs n ks l)) \<and> snd (dioph_ineq_dfs n ks l) ! k = i)"
Sigma_mset_mono: fixes A :: "'a multiset"   and C :: "'a multiset"   and B :: "'a \<Rightarrow> 'b multiset"   and D :: "'a \<Rightarrow> 'b multiset" assumes "A \<subseteq># C"   and "\<And>x. x \<in># A \<Longrightarrow> B x \<subseteq># D x" shows "Sigma_mset A B \<subseteq># Sigma_mset C D"
dghm_tdghm_comp_components_4: fixes \<HH> :: "V"   and \<NN> :: "V" shows "(\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>)\<lparr>NTDGDom\<rparr> = \<NN>\<lparr>NTDGDom\<rparr>"
smallest_accepting_rank_properties_4: fixes i :: "nat" assumes "smallest_accepting_rank = Some i" shows "infinite (succeed i)"
at_2: shows "atoms F \<subseteq> atoms_mset (\<Gamma>' + \<Delta>')"
tendsto: fixes g :: "'d \<Rightarrow> 'a"   and a :: "'a"   and F :: "'d filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. T (g x)) \<longlongrightarrow> T a) F"
IMPRIGHT_elim: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "<a>:M \<in> IMPRIGHT (B IMP C) (\<parallel>(B)\<parallel>) (\<parallel><C>\<parallel>) (\<parallel>(C)\<parallel>) (\<parallel><B>\<parallel>)" obtains x' :: "name" and a' :: "coname" and M' :: "trm" where "M = ImpR x'.<a'>.M' a" and "fic (ImpR x'.<a'>.M' a) a" and "\<forall>z P. x' \<sharp> (z, P) \<and> z:P \<in> \<parallel>(C)\<parallel> \<longrightarrow> x':M'{a':=z.P} \<in> \<parallel>(B)\<parallel>" and "\<forall>c Q. a' \<sharp> (c, Q) \<and> <c>:Q \<in> \<parallel><B>\<parallel> \<longrightarrow> <a'>:M'{x':=<c>.Q} \<in> \<parallel><C>\<parallel>"
dfs_invar_final: fixes \<Sigma>i :: "'d set"   and R :: "('d \<times> 'd) set" shows "\<forall>sc. sc \<in> wa_invar (dfs_algo \<Sigma>i R) \<and> sc \<notin> wa_cond (dfs_algo \<Sigma>i R) \<longrightarrow> fst (dfs_\<alpha> sc) = R\<^sup>* `` \<Sigma>i"
dvd_unit_imp_unit: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b"   and "dvd.dvd times b one" shows "dvd.dvd times a one"
simps_2: shows "ou_witness.b \<noteq> ou_witness.a"
map_rose_tree_F_const_def: fixes f1 :: "'c \<Rightarrow> 'e"   and f2 :: "'d \<Rightarrow> 'f" shows "map_rose_tree_F_const f1 f2 \<equiv> map_prod f1 (map f2)"
SUP_constant: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and c :: "'d"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>y. c) ` A) = (if A = {} then bot else c)"
silent_move_into_RedT_\<tau>_inv: fixes t :: "'t"   and x :: "'x"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x' :: "'x"   and m' :: "'m" assumes "silent_move t (x, shr s) (x', m')"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "wset s t = None" shows "\<tau>mredT s (redT_upd_\<epsilon> s t x' m')"
listToFun_def: fixes xs :: "('a \<times> 'b list) list" shows "listToFun xs \<equiv> foldr (\<lambda>(a, acts) M. concat (map (\<lambda>m. map (fun_upd m a) acts) M)) xs [\<lambda>_. undefined]"
le_rem_simps_1: fixes a :: "'c" shows "le_rem_s (LOC a) = a"
j_2: shows "\<forall>p. ps (coS t j) p \<noteq> None"
basic_cqe_conv1_5: fixes c :: "'c" shows "(\<exists>x. poly [:c:] x = (0::'c)) = (c = (0::'c))"
simps_1: fixes x :: "complex"   and y :: "complex" shows "Re (x + y) = Re x + Re y"
no_spoofing_def: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and rs :: "'b common_primitive rule list" shows "no_spoofing TYPE('a) ipassmt rs \<equiv> \<forall>iface\<in>dom ipassmt. \<forall>p. (common_matcher, in_doubt_allow),p\<lparr>p_iiface := iface_sel iface\<rparr>\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<longrightarrow> p_src p \<in> ipcidr_union_set (set (the (ipassmt iface)))"
nsmulextp_cong: fixes xs1 :: "'a multiset"   and ys1 :: "'a multiset"   and xs2 :: "'a multiset"   and ys2 :: "'a multiset"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" assumes "xs1 = ys1"   and "xs2 = ys2"   and "\<And>x x'. \<lbrakk>x \<in># ys1; x' \<in># ys2\<rbrakk> \<Longrightarrow> f x x' = g x x'" shows "nsmulextp f xs1 xs2 = nsmulextp g ys1 ys2"
return_cons_rule: fixes P :: "assn"   and Q :: "'b \<Rightarrow> assn"   and x :: "'b" assumes "P \<Longrightarrow>\<^sub>A Q x" shows "<P> return x <Q>"
num_eq_card_orbit: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and P :: "'c set" assumes "snd_sylow G p a m"   and "P \<in> group.subgroups_of_size G (p ^ a)" shows "group.subgroups_of_size G (p ^ a) = group_action.orbit G (group.conjugation_action G (p ^ a)) (group.subgroups_of_size G (p ^ a)) P"
Cd_XD_gfp_F_card: fixes d :: "'b" assumes "d \<in> ds" shows "card (Cd d (fst XD_XH)) \<le> card (Cd d (fst (gfp_F ds)))"
assms_2: fixes r :: "'a" assumes "Q r" shows "P r"
deny_pfun_def: fixes pf :: "'c \<Rightarrow> 'd option" shows "D\<^sub>p pf \<equiv> \<lambda>x. case pf x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>deny y\<rfloor>"
bebmovovissiffiddi: shows "b O (e \<union> b \<union> m \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse>) \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
zero_Nat_def: shows "(0::'a) = AbsNat 0"
subsp_3: shows "IrrFinGroupRepresentation G (\<cdot>) W"
closedin_subtopology_alt: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" shows "closedin (subtopology X U) S = (S \<in> (\<inter>) U ` Collect (closedin X))"
abs_eq: fixes x :: "char list" assumes "eq_onp (\<lambda>cs. \<forall>c\<in>set cs. \<not> digit7 c) x x" shows "asciis_of_literal (Abs_literal x) = map of_char x"
scop_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.scop prob \<equiv> {(xa, x). IFC_def.scopp prob xa x}"
length_alternating_list: fixes n :: "nat"   and s :: "'b"   and t :: "'b" shows "length (alternating_list n s t) = n"
A_Issues_B: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and NB :: "nat"   and evs :: "event list" assumes "Says A B (Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace>) \<in> set evs"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_shared" shows "A Issues B with Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace> on evs"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and x21 :: "'c"   and x22 :: "('c, 'b) tllist" shows "(case TCons x21 x22 of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
scaleRat_less2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and k :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less a b"   and "k < 0" shows "less (scaleRat k b) (scaleRat k a)"
i1: shows "i < length (status \<sigma> (f, length xs))"
Basis: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set"   and CC :: "'d set set" assumes "F \<in> A ensures B"   and "A - B \<in> insert {} CC" shows "(A, B) \<in> elt CC F"
fail_reachable_reverse: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM" assumes "well_formed M1"   and "well_formed M2"   and "productF M2 M1 FAIL PM"   and "FAIL \<in> reachable PM (initial PM)"   and "observable M2" shows "\<not> M1 \<preceq> M2"
single: fixes S :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "swapidseq_ext S n p"   and "a \<notin> S" shows "swapidseq_ext (insert a S) n p"
b_bounds: fixes b :: "real" assumes "b \<in> set bs" shows "b \<in> {0<..<1}"
sup_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<cdot> z = x \<cdot> z \<squnion> y \<cdot> z"
sets_borel_eq_count_space: shows "sets borel = sets (count_space UNIV)"
n_mult_def: fixes f :: "'c \<Rightarrow> 'a" shows "n_mult f = (\<lambda>n. [n] \<cdot> f n)"
subgame_V: fixes V' :: "'a set" shows "V\<^bsub>subgame V'\<^esub> \<subseteq> V"
finsum_cong2: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'a vec"   and g :: "'b \<Rightarrow> 'a vec" assumes "A = B"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i"   and "f \<in> B \<rightarrow> carrier_vec n" shows "finsum V f A = finsum V g B"
aux_3: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry"   and c :: "'d DBMEntry" assumes "\<not> a \<prec> b"   and "\<not> b \<prec> c"   and "a \<prec> c" shows "c = a"
fmdrop_lookup: fixes a :: "'e"   and m :: "('e, 'd) fmap" shows "fmlookup (fmdrop a m) a = None"
mult_strict_right_mono_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less b a"   and "less c zero" shows "less (times a c) (times b c)"
\<beta>: shows "LeftDerives1 (\<delta> @ \<alpha>) i r \<beta> \<and> ix < length (snd r) \<and> snd r ! ix = \<gamma> ! j \<and> LeftDerivationFix \<beta> (i + ix) D j \<gamma>"
elims_1: fixes x :: "'b status"   and y :: "bool" assumes "st_inv x = y" obtains ok :: "nat" and f :: "'b list" and f' :: "'b list" and r :: "'b list" and r' :: "'b list" where "x = Rev ok__ f f'__ r r'__" and "y = (length f + 1 = length r \<and> length f'__ = length r'__ \<and> ok__ \<le> length f'__)"   | ok :: "nat" and f' :: "'b list" and r' :: "'b list" where "x = App ok f' r'" and "y = (ok \<le> length f' \<and> length f' < length r')"   | "x = Idle" and "y"   | v :: "'b list" where "x = Done v" and "y"
sc_l_distr: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and a :: "'i"   and b :: "'i"   and m :: "'h" assumes "R module M"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "m \<in> carrier M" shows "(a \<plusminus>\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> b \<cdot>\<^sub>s\<^bsub>M\<^esub> m"
simps_4: fixes x :: "Vars.var"   and y :: "Vars.var"   and z :: "Vars.var"   and e :: "exp" assumes "atom x \<sharp> (y, z)" shows "(Lam [x]. e)[y::=z] = Lam [x]. e[y::=z]"
f_Exec_Stream_Acc_Output_causal: fixes n :: "nat"   and xs :: "'d message_af list"   and ys :: "'d message_af list"   and k :: "nat"   and output_fun :: "'f \<Rightarrow> 'e message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f" assumes "xs \<down> n = ys \<down> n" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c \<down> n = f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun ys c \<down> n"
zero_neq_one: shows "(0::'b) \<noteq> (1::'b)"
mat_eq: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list" assumes "mat nr nc m1"   and "mat nr nc m2" shows "(m1 = m2) = (\<forall>i<nc. \<forall>j<nr. m1 ! i ! j = m2 ! i ! j)"
decode_exp_shift_Suc: fixes e :: "nat"   and M :: "hmultiset" assumes "local.aligned\<^sub>d (Suc e) M" shows "local.decode e M = base * local.decode (Suc e) M"
a_consistent_if\<^sub>2: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and a' :: "Arity"   and as' :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, a' # as') (\<Gamma>, Bool b, Alts e1 e2 # S)" shows "a_consistent (ae, a', as') (\<Gamma>, if b then e1 else e2, S)"
genp_gen_eq: fixes P :: "'a set set"   and I :: "'a set" shows "genp (\<lambda>x. x \<in> P) (\<lambda>x. x \<in> I) = (\<lambda>x. x \<in> gen P I)"
RPC_action_defs_4: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCFail send rcv rst p \<equiv> \<not> $Calling rcv p \<and> Return send p #RPCFailure \<and> (rst!p)$ = #rpcA \<and> unchanged (caller rcv!p)"
resultant_sub: fixes p :: "'b poly"   and q :: "'b poly" shows "resultant p q = resultant_sub (degree p) (degree q) p q"
assms_1: fixes l :: "nat list"   and i :: "nat" assumes "l \<in> set subs1"   and "i \<in> set l" shows "i < n1"
vid_on_atI: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "vid_on A\<lparr>a\<rparr> = a"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "wake_up_status \<Rightarrow> 'a"   and wait_set_status :: "'b wait_set_status" shows "P (case wait_set_status of InWS x \<Rightarrow> f1 x | PostWS x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. wait_set_status = InWS x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. wait_set_status = PostWS x2 \<and> \<not> P (f2 x2))))"
C2: shows "C.commutative_square \<nu>'.cod.leg0 \<mu>'.cod.leg1 (Chn (\<nu>' \<bullet> \<nu>) \<cdot> \<p>\<^sub>1[\<nu>.dom.leg0, \<mu>.dom.leg1]) (Chn (\<mu>' \<bullet> \<mu>) \<cdot> \<p>\<^sub>0[\<nu>.dom.leg0, \<mu>.dom.leg1])"
append_butlast_last_id: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "butlast xs @ [List.last xs] = xs"
simps: fixes a1 :: "'v node"   and a2 :: "'v node" shows "edge' a1 a2 = (a1 = SOURCE \<and> a2 = Inner (source \<Delta>) \<or> (\<exists>x y. a1 = Inner x \<and> a2 = Inner y \<and> edge \<Delta> x y))"
simps: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and A :: "atomUni" shows "quadraticSubUni a b c d A = (if aEvalUni A ((a + b * sqrt c) / d) then TrueFUni else FalseFUni)"
dual_star_pres_disj: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> disjunctive.disjunctive sup sup times" shows "dual_star x \<in> disjunctive.disjunctive sup sup times"
map_add_dom_app_simps_1: fixes m :: "'b"   and l2 :: "'b \<Rightarrow> 'c option"   and l1 :: "'b \<Rightarrow> 'c option" assumes "m \<in> dom l2" shows "(l2 \<Oplus> l1) m = l2 m"
mpt_factor_def: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" shows "mpt_factor proj M2 T2 = (proj \<in> measure_preserving M M2 \<and> (AE x in M. proj (T x) = T2 (proj x)) \<and> mpt M2 T2)"
is_real_mult_real: fixes a :: "complex"   and b :: "complex" assumes "is_real a"   and "a \<noteq> 0" shows "is_real b = is_real (a * b)"
abs_less_imp_sq_less: fixes x :: "real"   and y :: "real"   and z :: "real"   and w :: "real" assumes "\<bar>x - y\<bar> < \<bar>z - w\<bar>" shows "(x - y)\<^sup>2 < (z - w)\<^sup>2"
fs0_gso0: assumes "0 < length fs" shows "fs ! 0 = gso 0"
fexpr_in_carrier: fixes xs :: "'a list"   and e :: "fexpr" assumes "in_carrier xs"   and "nonzero xs (Cond e)" shows "feval xs e \<in> carrier R"
assms_6: shows "sig_red_zero (\<prec>\<^sub>t) G p"
cont_Ch_L: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and s :: "'state" shows "local.cont (Ch ch c1 c2) s 0 = c1"
max_mono: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c"   and z :: "'c" assumes "class.non_strict_order less_eq"   and "less_eq y x" shows "less_eq (ord.max less_eq z y) (ord.max less_eq z x)"
Alm_all_def: fixes P :: "'b \<Rightarrow> bool" shows "Alm_all P = (\<not> (\<exists>\<^sub>\<infinity>x. \<not> P x))"
nonempty_domain_enumerable: fixes i :: "nat"   and x\<^sub>0 :: "nat" assumes "eval r_phi [i, x\<^sub>0] \<down>" shows "\<exists>g. recfn 1 g \<and> Partial_Recursive.total g \<and> (\<forall>x. (eval r_phi [i, x] \<down>) = (\<exists>y. eval g [y] \<down>= x))"
a_subgroup: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "additive_subgroup H G" shows "subgroup H (add_monoid G)"
arev_dom: fixes G :: "('a, 'b) pre_digraph"   and arev :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bidirected_digraph G arev" shows "(a \<in> arcs G) = (arev a \<noteq> a)"
check_ipresIGSwap: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGSwap check (errMOD MOD) MOD"
monoid_axioms: shows "monoid (AND) (- (1::'c))"
ennreal_of_nat_neq_top: fixes i :: "nat" shows "of_nat i \<noteq> top"
chain: shows "\<forall>i. (f i, f (Suc i)) \<in> r"
3: shows "codomain_block (make_vert_block n) = int n"
associated_polynomials_imp_same_length: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subring K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "p \<sim>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> q" shows "length p = length q"
exp_diff: fixes x :: "'a"   and y :: "'a" shows "exp (x - y) = exp x / exp y"
RedRInit: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and v :: "val"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, i)\<rfloor>"   and "sh' = sh(C \<mapsto> (sfs, Done))"   and "C' = last (C # Cs)" shows "P \<turnstile> \<langle>RI (C,Val v) ; Cs \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh'),b\<rangle>"
induct: fixes x1 :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and x2 :: "typ"   and P :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option) \<Rightarrow> typ \<Rightarrow> bool" assumes "wf_type x1 x2"   and "\<And>\<Sigma> \<kappa> Ts. \<lbrakk>type_arity \<Sigma> \<kappa> = Some (length Ts); \<forall>T\<in>set Ts. wf_type \<Sigma> T \<and> P \<Sigma> T\<rbrakk> \<Longrightarrow> P \<Sigma> (Ty \<kappa> Ts)"   and "\<And>\<Sigma> S a. wf_sort (subclass (osig \<Sigma>)) S \<Longrightarrow> P \<Sigma> (Tv a S)" shows "P x1 x2"
t1_def: shows "t1 = punit.lt (rep_list p)"
sel_2: fixes x21 :: "tree"   and x22 :: "tree" shows "rtree (Branching x21 x22) = x22"
coprime_m_def: fixes f :: "int poly"   and g :: "int poly" shows "p.coprime_m f g = (\<forall>h. p.dvdm h f \<longrightarrow> p.dvdm h g \<longrightarrow> p.dvdm h 1)"
host_shadow_root_rel_def: shows "host_shadow_root_rel \<equiv> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_host_shadow_root_rel get_shadow_root"
poly_add_def: fixes p :: "'a poly"   and q :: "'a poly" shows "poly_add p q = resultant (poly_x_minus_y p) (poly_lift q)"
take_zip: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'e list" shows "take n (xs || ys) = take n xs || take n ys"
cases: fixes x :: "nat list" obtains "x = []"   | xa :: "nat" and xs :: "nat list" where "x = xa # xs"
simps_7: fixes v :: "'c match_expr"   and vb :: "char list"   and rs :: "'c rule list" shows "process_ret (Rule v (Call vb) # rs) = Rule v (Call vb) # process_ret rs"
irreflexiveDecisionLess: fixes x :: "'b \<times> bool" shows "(x, x) \<notin> decisionLess"
indis_refl: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and s :: "'b" assumes "PL_Indis cval indis" shows "(s, s) \<in> indis"
in_cont_mkTree: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and S :: "'b set"   and s :: "'b"   and rs :: "'a stream"   and t' :: "('b \<times> 'a) tree" assumes "RuleSystem eff rules S"   and "s \<in> S"   and "i.fair rules rs"   and "t' |\<in>| cont (RuleSystem_Defs.mkTree eff rs s)" shows "\<exists>sl' s'. s' \<in> S \<and> eff (shd (RuleSystem_Defs.trim eff rs s)) s sl' \<and> s' |\<in>| sl' \<and> t' = RuleSystem_Defs.mkTree eff (stl (RuleSystem_Defs.trim eff rs s)) s'"
c: shows "Y \<in> obj\<^bsub>C\<^esub>"
word_split_def: fixes w :: "'d word" shows "word_split w = (ucast (drop_bit LENGTH('c) w), ucast w)"