hcju/isabelleps huggingface dataset:
irreducibleD: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "irreducible G a"   and "properfactor G b a"   and "b \<in> carrier G" shows "b \<in> Units G"
hyps_1: fixes qs :: "nat list" assumes "k = length qs" shows "\<exists>N. partn_lst {..<N} qs r"
simps_9: fixes f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "merge_guards (Guard f g c) = (let c' = merge_guards c in if is_Guard c' then let (f', g', c'') = dest_Guard c' in if f = f' then Guard f (g \<inter> g') c'' else Guard f g (Guard f' g' c'') else Guard f g c')"
abs_eq: fixes x :: "'b ptrm" shows "Fst (abs_trm x) = abs_trm (PFst x)"
min_list_param_minimal: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and z :: "'a" assumes "transp rel"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> rel x y \<or> rel y x"   and "z \<in> set xs" shows "rel (min_list_param rel xs) z"
distinct_map_fstD: fixes xs :: "('b \<times> 'c) list"   and x :: "'b"   and y :: "'c"   and z :: "'c" assumes "distinct (map fst xs)"   and "(x, y) \<in> set xs"   and "(x, z) \<in> set xs" shows "y = z"
S3: fixes v1 :: "'a"   and v2 :: "'a" assumes "v1 \<in> insert x S"   and "v2 \<in> insert x S" shows "{v1, v2} \<notin> E"
card_insert_if: fixes A :: "'c set"   and x :: "'c" assumes "finite A" shows "card (insert x A) = (if x \<in> A then card A else Suc (card A))"
degeneralize_nodes_finite: fixes A :: "('b, 'a) ngba" shows "finite (NBA.nodes (degeneralize A)) = finite (NGBA.nodes A)"
inclusion: fixes f :: "'a \<times> 'a" assumes "B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f)" shows "B.VxV.arr f"
simps_5: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" shows "clearQuantifiers (Or \<phi> \<psi>) = or (clearQuantifiers \<phi>) (clearQuantifiers \<psi>)"
prems_3: shows "set (a # Es) \<subseteq> E"
homeomorphic_subspaces: fixes S :: "'a set"   and T :: "'b set" assumes "subspace S"   and "subspace T"   and "dim S = dim T" shows "S homeomorphic T"
prefixes_length_mono': fixes j :: "nat"   and t :: "nat"   and d :: "nat" shows "e_length (local.prefixes j t) \<le> e_length (local.prefixes j (t + d))"
Rats_eq_range_nat_to_rat_surj: shows "\<rat> = range nat_to_rat_surj"
lift_NF: fixes t :: "dB"   and i :: "nat" assumes "NF t" shows "NF (lift t i)"
ineq_wf_dfa: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "wf_dfa (ineq_dfa n ks l) n"
ltln_weak_to_strong_4: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi>"   and "w \<Turnstile>\<^sub>n F\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi>"
geometric_series_def: fixes n :: "nat"   and a :: "'a"   and b :: "'a" shows "geometric_series n a b = to_fun (UP_root_div (monom P \<one> (Suc n) \<ominus>\<^bsub>UP R\<^esub> monom P (b [^] Suc n) 0) b) a"
map_snd_zip_take: fixes xs :: "'d list"   and ys :: "'c list" shows "map snd (xs || ys) = take (min (length xs) (length ys)) ys"
fle_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fle a b = (fcompare a b = Lt \<or> fcompare a b = Eq)"
filterlim_int_of_nat_at_topD: fixes f :: "int \<Rightarrow> 'a"   and F :: "'a filter" assumes "LIM x sequentially. f (int x) :> F" shows "filterlim f F at_top"
add_strict_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a < b"   and "c < d" shows "a + c < b + d"
Nats: fixes c :: "real" assumes "c \<in> \<nat>" shows "c \<in> P"
unique_connecting_path_properties_2: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "v \<leadsto>\<^bsub>G\<^esub> w \<noteq> []"
no_move1_to_no_move2: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "\<And>tl1 s1'. \<not> trsys1 s1 tl1 s1'" shows "\<exists>s2'. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>*\<^sup>* s2 s2' \<and> (\<forall>tl2 s2''. \<not> trsys2 s2' tl2 s2'') \<and> bisim s1 s2'"
induct: fixes x :: "atom"   and T :: "atom set"   and H :: "chmsg set"   and P :: "atom \<Rightarrow> bool" assumes "x \<in> extr T H"   and "\<And>At. At \<in> T \<Longrightarrow> P At"   and "\<And>c A B M At. \<lbrakk>StatCh c A B (Msg M) \<in> H; At \<in> set M; confid \<notin> c \<or> A \<in> bad \<or> B \<in> bad\<rbrakk> \<Longrightarrow> P At"   and "\<And>c K M At. \<lbrakk>DynCh c K (Msg M) \<in> H; At \<in> set M; confid \<notin> c \<or> aKey K \<in> extr T H \<and> P (aKey K)\<rbrakk> \<Longrightarrow> P At" shows "P x"
multiplicity_prime_prime: fixes p :: "'a"   and q :: "'a" assumes "prime p"   and "prime q" shows "multiplicity p q = (if p = q then 1 else 0)"
shows_nl_def: shows "shows_nl = shows CHR ''\<newline>''"
param_dropWhile: fixes a :: "('a \<times> 'b) set" shows "(dropWhile, dropWhile) \<in> (a \<rightarrow> bool_rel) \<rightarrow> \<langle>a\<rangle>list_rel \<rightarrow> \<langle>a\<rangle>list_rel"
update_convs_8: fixes return' :: "t list option \<Rightarrow> t list option"   and types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "return_update return' \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return' return, \<dots> = more\<rparr>"
is_pkt_handler_num_leq_no: fixes i :: "nat" shows "optoy i \<Turnstile> (otherwith nos_inc {i} (orecvmsg msg_ok), other nos_inc {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<sigma>, l). l \<in> {PToy-:6..PToy-:10} \<longrightarrow> num (\<sigma> i) \<le> no (\<sigma> (sid (\<sigma> i))))"
assms_2: shows "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"
fv_regex_commute: fixes r :: "'a Regex.regex"   and x :: "'b"   and fv :: "'a \<Rightarrow> 'b set"   and g :: "'b \<Rightarrow> 'c"   and fv' :: "'a \<Rightarrow> 'c set" assumes "\<And>z. z \<in> regex.atms r \<Longrightarrow> (x \<in> fv z) = (g x \<in> fv' z)" shows "(x \<in> Regex.fv_regex fv r) = (g x \<in> Regex.fv_regex fv' r)"
abs_summable_in_conv_radius: fixes z :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "ereal (norm z) < conv_radius f" shows "summable (\<lambda>n. norm (f n * z ^ n))"
sign_simps_29: fixes am :: "'a"   and bm :: "'a" shows "am + bm = bm + am"
TIP: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "let PP = \<lambda>t. subst P t xx in isTrue (PP \<langle>\<phi>\<rangle>)" shows "prv \<phi>"
F_def: shows "F \<equiv> {accn i |i. i < num_acc}"
alphaAbs_qAbs_iff_all_equal_or_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<forall>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
eval_singletonI: fixes A :: "'a pred"   and default :: "unit \<Rightarrow> 'a" assumes "\<exists>!x. eval A x" shows "eval A (singleton default A)"
weakly_step_consistent_def: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" shows "weakly_step_consistent P D R \<equiv> \<forall>u\<in>range D. \<forall>xs ys x. (xs, ys) \<in> R u \<inter> R (D x) \<and> x \<in> next_events P xs \<inter> next_events P ys \<longrightarrow> (xs @ [x], ys @ [x]) \<in> R u"
set_rel_id_simp: shows "\<langle>Id\<rangle>set_rel = Id"
bij_betw_swap_iff: fixes x :: "'a"   and A :: "'a set"   and y :: "'a"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "x \<in> A"   and "y \<in> A" shows "bij_betw (Fun.swap x y f) A B = bij_betw f A B"
update_convs_3: fixes c_imp' :: "('a \<Rightarrow> 'b zmultiset) \<Rightarrow> 'a \<Rightarrow> 'b zmultiset"   and c_work :: "'a \<Rightarrow> 'b zmultiset"   and c_pts :: "'a \<Rightarrow> 'b zmultiset"   and c_imp :: "'a \<Rightarrow> 'b zmultiset"   and more :: "'c" shows "c_imp_update c_imp' \<lparr>c_work = c_work, c_pts = c_pts, c_imp = c_imp, \<dots> = more\<rparr> = \<lparr>c_work = c_work, c_pts = c_pts, c_imp = c_imp' c_imp, \<dots> = more\<rparr>"
poly_mult_lead_coeff_aux: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2"   and "p1 \<noteq> []"   and "p2 \<noteq> []" shows "local.coeff (poly_mult p1 p2) (degree p1 + degree p2) = lead_coeff p1 \<otimes> lead_coeff p2"
B_authenticates_A: fixes servK :: "nat"   and A :: "agent"   and T3 :: "nat"   and evs :: "event list"   and B :: "agent"   and Ts :: "nat" assumes "Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbV" shows "\<exists>ST. Says A B \<lbrace>ST, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"
subformulas\<^sub>\<mu>\<^sub>\<nu>_card: fixes \<phi> :: "'b ltln" shows "card (subformulas\<^sub>\<mu> \<phi> \<union> subformulas\<^sub>\<nu> \<phi>) = card (subformulas\<^sub>\<mu> \<phi>) + card (subformulas\<^sub>\<nu> \<phi>)"
of_nat_simps_2: fixes m :: "nat" shows "of_nat (Suc m) = (1::'a) + of_nat m"
hupd_lupd: fixes a :: "loc"   and obj :: "obj"   and x :: "vname"   and y :: "val"   and s :: "state" shows "hupd(a\<mapsto>obj) (lupd(x\<mapsto>y) s) = lupd(x\<mapsto>y) (hupd(a\<mapsto>obj) s)"
len_inf_elem: fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "(a, b) \<in> set (arcs i j xs)"   and "M a b = \<infinity>" shows "len M i j xs = \<infinity>"
not_in_net_ips_fst_init_missing: fixes i :: "nat"   and \<sigma> :: "((state \<times> 'a) \<times> 'b) net_state" assumes "i \<notin> net_ips \<sigma>" shows "fst (initmissing (netgmap fst \<sigma>)) i = aodv_init i"
A: fixes s :: "'d" shows "c_ipurge I (c_dom D) u (c_tr step out s (c_ipurge I D u xs)) = c_tr step out s (c_ipurge I D u xs)"
connected_def: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open" shows "topological_space.connected open S = (\<nexists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
perfect_hiding: fixes \<A> :: "('a \<times> nat \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'c) spmf) \<times> ('a \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "okamoto_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<A>"
inside_Un_outside: fixes S :: "'a set" shows "inside S \<union> outside S = - S"
less_imp_not_eq2: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "x \<prec> y" shows "(y = x) = False"
inj_imp_inv_eq: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" assumes "inj f"   and "\<forall>x. f (g x) = x" shows "inv f = g"
lm104: fixes N :: "'c set"   and G :: "'d list" assumes "0 < card N"   and "distinct G" shows "\<Union> {injections P N |P. P \<in> all_partitions (set G)} = \<Union> (set (map (\<lambda>l. set (injections_alg l N)) (all_partitions_list G)))"
BIT_bin_simps_1: fixes k :: "num" shows "numeral k BIT False = numeral (num.Bit0 k)"
Chains_init_seg_of_Union: fixes R :: "('b \<times> 'b) set set"   and r :: "('b \<times> 'b) set" assumes "R \<in> Chains init_seg_of"   and "r \<in> R" shows "r initial_segment_of \<Union> R"
run_option_def: fixes optionT :: "('b, 'a) optionT" shows "run_option optionT = (case optionT of OptionT x \<Rightarrow> x)"
disjI2: fixes s :: "Store"   and y :: "Value"   and x :: "Value" assumes "\<And>l. s\<turnstile> l reachable_from y \<Longrightarrow> \<not> s\<turnstile> l reachable_from x" shows "StoreProperties.disj x y s"
fset: fixes x :: "'a fset" shows "fset x \<in> {A. finite A}"
delete_def: fixes a :: "'b"   and t :: "'b bro" shows "delete a t = tree (del a t)"
v0_conforms: assumes "v \<in> VV p**" shows "\<sigma>' v = comp.w0"
collapse: fixes item :: "item" shows "Item (item_rule item) (item_dot item) (item_origin item) (item_end item) = item"
vec_first_def: fixes v :: "'b vec"   and n :: "nat" shows "vec_first v n \<equiv> vec n (($) v)"
AE_iff_null_sets: fixes P :: "'a \<Rightarrow> bool" shows "almost_everywhere M P = ({x \<in> space M. \<not> P x} \<in> null_sets M)"
discrL_def: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "discrL cl \<equiv> Ball (set cl) discr"
update_convs_3: fixes lim' :: "nat \<Rightarrow> nat"   and arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'b" shows "lim_update lim' \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays, refs = refs, lim = lim' lim, \<dots> = more\<rparr>"
sum_list_zip_map: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "distinct xs" shows "sum_list (map2 f xs (map g xs)) = (\<Sum>x\<in>set xs. f x (g x))"
assms: shows "incseq (\<lambda>n x. f n (x, y))"
exec_1I: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>" shows "P \<turnstile> \<sigma> -jvm\<rightarrow>\<^sub>1 \<sigma>'"
bind_return_pmf': fixes N :: "'d pmf" shows "N \<bind> return_pmf = N"
left_neutral: fixes a :: "'k" shows "\<top> \<sqinter> a = a"
torder_onI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "porder_on A r"   and "total_onP A r" shows "torder_on A r"
row_1: fixes i :: "nat" assumes "i < nr1" shows "row (four_block_mat A1 B1 C1 D1) i = row A1 i @\<^sub>v row B1 i"
assms_2: shows "inner_aforms' p XS (map pdevs_of_real rs) = Some R"
preorder_on_axioms: shows "Order_Predicates.preorder_on (f ` carrier) (map_relation (inv f) le)"
isMaxim_def: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and b :: "'d" assumes "wo_rel r" shows "wo_rel.isMaxim r A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (a, b) \<in> r)"
exhaust: fixes y :: "('a, 'b) ta_rule" obtains x1 :: "'a" and x2 :: "'b" and x3 :: "'a list" where "y = x1 \<rightarrow> x2 x3"
set_integrable_mult_right: fixes a :: "'b"   and M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "a \<noteq> (0::'b) \<Longrightarrow> set_integrable M A f" shows "set_integrable M A (\<lambda>t. a * f t)"
hyps_3: shows "\<forall>cn. mon_w fg w \<inter> mon_c fg cn = {} \<longrightarrow> (\<exists>csp. c' = csp + ch \<and> mon_c fg csp = {} \<and> ((sh, cn), w, s', csp + cn) \<in> trcl (trss fg))"
less_asym': fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and P :: "bool" assumes "ord_strict_p a b"   and "ord_strict_p b a" shows "P"
A_incr: fixes bs :: "bool list" shows "BinCounter.T_incr bs + BinCounter.p_incr (BinCounter.incr bs) - BinCounter.p_incr bs = 2"
subsumers_conv: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "subsumers subs = {v'. \<exists>v. (v, v') \<in> subs}"
simps_5: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "next \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = next"
assign_def: fixes f :: "'d control_state_scheme \<Rightarrow> 'd control_state_scheme" shows "assign f = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else Some ((), f \<sigma>))"
is_process_def: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" shows "is_process P = (([], {}) \<in> FAILURES P \<and> (\<forall>s X. (s, X) \<in> FAILURES P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<in> FAILURES P \<longrightarrow> (s, {}) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, Y) \<in> FAILURES P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, X) \<in> FAILURES P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> FAILURES P) \<longrightarrow> (s, X \<union> Y) \<in> FAILURES P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> FAILURES P \<longrightarrow> (s, X - {tick}) \<in> FAILURES P) \<and> (\<forall>s t. s \<in> DIVERGENCES P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> DIVERGENCES P) \<and> (\<forall>s X. s \<in> DIVERGENCES P \<longrightarrow> (s, X) \<in> FAILURES P) \<and> (\<forall>s. s @ [tick] \<in> DIVERGENCES P \<longrightarrow> s \<in> DIVERGENCES P))"
rename_Constrains: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "bij h" shows "(rename h F \<in> h ` A Co h ` B) = (F \<in> A Co B)"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir1: fixes fs :: "nat list" assumes "0 < length fs" shows "FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m 1 fs \<sqsubseteq> FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m 0 (fs ! 0)"
f1: fixes a :: "'a" shows "ad (ad (ad a) \<cdot> ad (ad a)) = ad a"
find_nu_measure_decreases: fixes \<nu> :: "nat" assumes "seq (\<nu> + \<nu>) \<noteq> seq \<nu>"   and "\<nu> \<le> nu_witness" shows "(Suc \<nu>, \<nu>) \<in> find_nu_measure"
get_option_def: fixes get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> ('d, 'e) optionT" shows "get_option get f = OptionT (get (\<lambda>s. run_option (f s)))"
measurable_funpow_T: fixes n :: "nat"   and m :: "nat" shows "T ^^ n \<in> tail_algebra (m + n) \<rightarrow>\<^sub>M tail_algebra m"
ordered_resolvent_is_resolvent: shows "less_restrictive resolvent ordered_resolvent"
not_sinvar_off: assumes "\<not> c_sinvar m \<lparr>nodes = V, edges = E\<rparr>" shows "c_offending_flows m \<lparr>nodes = V, edges = E\<rparr> \<noteq> {}"
assms: shows "run_one_step d i (s, vs, ves, $Select) = (s', vs', res)"
word_ao_dist2: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "x && (y || z) = x && y || x && z"
set_awalk_verts_cas: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.cas G u p v" shows "set (pre_digraph.awalk_verts G u p) = {u} \<union> set (map (tail G) p) \<union> set (map (head G) p)"
neq_local_nested: fixes n :: "nat"   and i :: "nat"   and lvs :: "v list"   and es :: "e list" shows "[Local n i lvs es] \<noteq> es"
merge_poly_def: shows "merge_poly = PAC_Checker_Init.merge (\<lambda>a b. fst a \<le> fst b)"
ent_pure_post_iff: fixes P :: "assn"   and b :: "bool" shows "(P \<Longrightarrow>\<^sub>A emp * \<up> b) = ((\<forall>h. h \<Turnstile> P \<longrightarrow> b) \<and> (P \<Longrightarrow>\<^sub>A emp))"
g1g: shows "g1 \<in> {{x} \<times> I |x. x \<in> bl}"
istopology_base: fixes P :: "'c set \<Rightarrow> bool" assumes "\<And>S T. \<lbrakk>P S; P T\<rbrakk> \<Longrightarrow> P (S \<inter> T)" shows "istopology (arbitrary union_of P)"
ordinal_exp_not_0: fixes x :: "ordinal"   and y :: "ordinal" shows "(0 < x ** y) = (0 < x \<or> y = 0)"
circ_unfold_sum: fixes y :: "'a" shows "top = top \<squnion> top \<sqinter> y \<sqinter> top"
oid_is_typerepr: shows "\<tau> \<Turnstile> X->forAll\<^sub>S\<^sub>e\<^sub>t(a|not (x \<triangleq> a))"
simps_3: fixes e :: "'b" shows "alist_and' [Neg e] = MatchNot (Match e)"
cet_mult_real: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL" shows "\<Gamma> \<turnstile>\<^sub>c e1 *\<^sub>c e2 : REAL"
Src_Trg: fixes t :: "'c local.term" assumes "Arr t" shows "Src (Trg t) = Trg t"
ldropWhile_lappend: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist"   and ys :: "'b llist" shows "ldropWhile P (lappend xs ys) = (if \<exists>x\<in>lset xs. \<not> P x then lappend (ldropWhile P xs) ys else if lfinite xs then ldropWhile P ys else LNil)"
inB: shows "insertion (nth_default 0 ((xs' @ x # xs)[var := - C / B])) b = B"
qfree_qe_dlo\<^sub>1: fixes as :: "atom list" shows "qfree (qe_dlo\<^sub>1 as)"
greatest_postfixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_postfixpoint f" shows "is_greatest_postfixpoint f (p\<nu> f)"
map_upds_Some_expand: fixes tab :: "'b \<Rightarrow> 'a option"   and vn :: "'b"   and z :: "'a"   and xs :: "'b list"   and ys :: "'a list" assumes "tab vn = Some z" shows "\<exists>z. (tab(xs [\<mapsto>] ys)) vn = Some z"
1: shows "D = P1 ** A ** Q1 \<and> invertible P1 \<and> invertible Q1 \<and> isDiagonal D"
indRelLp_indRelL_eq: shows "indRelLp = indRelLinfix"
hom_lead_coeff: fixes p :: "real poly" shows "lead_coeff (map_poly of_real p) = of_real (lead_coeff p)"
ANR_openin: fixes T :: "'a set"   and S :: "'a set" assumes "ANR T"   and "openin (top_of_set T) S" shows "ANR S"
minVertex_def: fixes f :: "face" shows "minVertex f \<equiv> ListAux.min_list (vertices f)"
interior_of_openin: fixes X :: "'b topology"   and S :: "'b set" assumes "openin X S" shows "X interior_of S = S"
cSup_atMost: 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 x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Sup (ord.atMost less_eq x) = x"
prems_1: shows "\<rho> \<in> carrier_mat d d"
ndet_distr: fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and Q :: "'c \<times> 'c \<Rightarrow> bool"   and R :: "'c \<times> 'c \<Rightarrow> bool" shows "(P \<or> Q \<or> R) = ((P \<or> Q) \<or> P \<or> R)"
convergent_powser'_ln: shows "convergent_powser' (msllist_of_msstream ln_series_stream) (\<lambda>x. ln (1 + x))"
typeof_simps_6: fixes t :: "Arraytype"   and a :: "nat" shows "\<tau> arrV t a = ArrT t"
b: shows "b = pp_of_term u - lp g1 + punit.lt (rep_list g1) - punit.lt (rep_list h)"
invertible: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "group G composition unit"   and "u \<in> G" shows "monoid.invertible G composition unit u"
sets_distinct1: fixes n :: "int"   and m :: "int" assumes "n \<noteq> m" shows "{(a, b). a = n} \<noteq> {(a, b). a = m}"
ed_ou_eqs_3: fixes X :: "'a set" assumes "ed_ou X" shows "I \<circ> K = K \<circ> I"
test_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "test x"   and "test y"   and "test z" shows "(z \<le> x \<sqinter> y) = (z \<le> x \<and> z \<le> y)"