hcju/isabelleps huggingface dataset:
distinct_vertices12: fixes vertex_1 :: "'a"   and vertex_2 :: "'a"   and vertex_3 :: "'a" assumes "class.vertex vertex_1 vertex_2 vertex_3" shows "vertex_1 \<noteq> vertex_2"
exhaust: fixes y :: "'a sequent" obtains x1 :: "'a form multiset" and x2 :: "'a form multiset" where "y = ( x1 \<Rightarrow>* x2)"
inj_distr: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_inj z" shows "(x \<cdot> y) ; z = x ; z \<cdot> y ; z"
uint32_simps_3: fixes x :: "uint32"   and y :: "uint32" shows "int_of_uint32 (x - y) = (int_of_uint32 x - int_of_uint32 y) mod 4294967296"
abelian_monoidE_1: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "abelian_monoid R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
simps: fixes a1 :: "'g"   and a2 :: "'a"   and a3 :: "'h + 'b" shows "(((a1, a2), a3) \<in> Rel_OR) = ((\<exists>x0 w0 x1. a1 = x0 \<and> a2 = x1 \<and> a3 = Inl w0 \<and> (x0, w0) \<in> Rel0 \<and> x1 \<in> valid_pub1) \<or> (\<exists>x1 w1 x0. a1 = x0 \<and> a2 = x1 \<and> a3 = Inr w1 \<and> (x1, w1) \<in> Rel1 \<and> x0 \<in> valid_pub0))"
prefix_primrec: shows "prefix = rec_list (\<lambda>xs. True) (\<lambda>x xs xsa ys. case ys of [] \<Rightarrow> False | y # ys \<Rightarrow> x = y \<and> xsa ys)"
B: shows "four_block_mat B1 UR LL B2 \<in> carrier_mat (n + m) (n + m)"
Abs_fps_inject: fixes x :: "'c list \<Rightarrow> 'd"   and y :: "'c list \<Rightarrow> 'd" assumes "x \<in> {f. True}"   and "y \<in> {f. True}" shows "(Abs_fps x = Abs_fps y) = (x = y)"
P: shows "P \<in> carrier_mat CARD('nr mod_ring) CARD('nr mod_ring)"
finite_Collapse_state: fixes Q :: "hf set" assumes "Q \<in> states M // eq_right_lang" shows "finite Q"
matrix_inv_Gauss: fixes A :: "(('b, 'c) vec, 'c) vec"   and P :: "(('b, 'c) vec, 'c) vec" assumes "invertible A"   and "Gauss_Jordan A = P ** A" shows "matrix_inv A = P"
trace_rest_aux_append: fixes rest :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme"   and p :: "'a spmf"   and tr :: "('c \<times> 'e \<times> 'b list + 'd \<times> 'f \<times> 'b list) list"   and tr' :: "('c \<times> 'e \<times> 'b list + 'd \<times> 'f \<times> 'b list) list" shows "trace_rest_aux rest p (tr @ tr') = trace_rest_aux rest (trace_rest_aux rest p tr) tr'"
signed_take_bit_int_less_self_iff: fixes n :: "nat"   and k :: "int" shows "(signed_take_bit n k < k) = (2 ^ n \<le> k)"
raw_has_prod_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'e"   and M :: "nat"   and p :: "'e"   and N :: "nat" assumes "raw_has_prod f M p"   and "M \<le> N" obtains q :: "'e" where "raw_has_prod f N q"
A_B_step: fixes a :: "'a"   and b :: "'a"   and a' :: "'b" assumes "A a b"   and "PA a"   and "PB a'"   and "sim a a'" shows "\<exists>b'. B a' b' \<and> sim b b'"
iso_dghm_ArrMap_inv_dghm_ArrMap_app: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>inv_dghm \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>\<rparr> = f"
dagger_simr: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "z \<cdot> x \<le> y \<cdot> z" shows "z \<cdot> x\<^sup>\<star> \<le> y\<^sup>\<star> \<cdot> z"
assms: shows "xs \<subseteq>\<^sub>\<circ> ys"
\<omega>consistentStd1_implies_consistent: assumes "B.\<omega>consistentStd1" shows "B.consistent"
mult_inv_same_comm: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "a \<in> Units G" shows "inv (a [^] m) \<otimes> inv (a [^] n) = inv (a [^] n) \<otimes> inv (a [^] m)"
plus_permutation_abs_eq: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "bij f"   and "bij g" shows "Abs_permutation f + Abs_permutation g = Abs_permutation (f \<circ> g)"
wpcarr: shows "wp \<in> carrier (relative_homology_group (int n) (nsphere n) (upper n))"
simps_1: fixes d :: "'a" shows "cast_Dyn (OpDyn d) = Some d"
rotate_is_Nil_conv: fixes n :: "nat"   and xs :: "'g list" shows "(rotate n xs = []) = (xs = [])"
upd_incoming_def: fixes n :: "('a, 'c) node_scheme"   and ns :: "('a, 'b) node_scheme set" shows "upd_incoming n ns \<equiv> upd_incoming_f n ` ns"
power_Suc_le_self: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq a one" shows "less_eq (power.power one times a (Suc n)) a"
untaints_def: fixes ts :: "SINVAR_TaintingTrusted.taints" shows "untaints ts \<equiv> taints_raw.untaints_raw (taints.raw_of_taints ts)"
emptyI: fixes e :: "'e"   and D :: "'e set"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "e \<in> D" shows "({}, e) \<in> foldSetD D f e"
wls_freshAbs_substAbs: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and y :: "'var" assumes "wlsAbs (us, s) A"   and "wls (asSort ys) Y" shows "freshAbs zs z (A $[Y / y]_ys) = ((zs = ys \<and> z = y \<or> freshAbs zs z A) \<and> (freshAbs ys y A \<or> fresh zs z Y))"
sel_3: fixes xs :: "'a llist"   and b :: "'b" assumes "\<not> lnull xs" shows "ttl (tllist_of_llist b xs) = (case xs of LCons x xs' \<Rightarrow> tllist_of_llist b xs')"
can_syncE: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2"   and LT :: "('l + 't + 't) set" assumes "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>2" obtains ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" and x' :: "'x2" and m' :: "'m2" where "r2.r_syntax t x m ta x' m'" and "LT = collect_waits ta"
snd_comp_apfst: fixes f :: "'d \<Rightarrow> 'f" shows "snd \<circ> apfst f = snd"
Parallel_length_post_PStar: fixes Ts :: "('b ann_com option \<times> 'b set) list"   and s :: "'b"   and R' :: "'b com"   and t :: "'b" assumes "(Parallel Ts, s) -P*\<rightarrow> (R', t)" shows "\<exists>Rs. R' = Parallel Rs \<and> length Rs = length Ts \<and> (\<forall>i<length Ts. OG_Hoare.post (Ts ! i) = OG_Hoare.post (Rs ! i))"
gSwapGVar_def: fixes MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'g, 'h, 'i) Recursion.model_scheme" shows "gSwapGVar MOD \<equiv> \<forall>zs z1 z2 xs x. gSwap MOD zs z1 z2 (Var xs x) (gVar MOD xs x) = gVar MOD xs (x @xs[z1 \<and> z2]_zs)"
fls_compose_power_fps_to_fls: fixes d :: "nat"   and f :: "'b fps" assumes "0 < d" shows "fls_compose_power (fps_to_fls f) d = fps_to_fls (f oo fps_X ^ d)"
prime_nat_iff: fixes n :: "nat" shows "prime n = (1 < n \<and> (\<forall>m. m dvd n \<longrightarrow> m = 1 \<or> m = n))"
inverse_arrows_sym: fixes f :: "'a1"   and g :: "'a1" shows "C1.inverse_arrows f g = C1.inverse_arrows g f"
Compl_disjoint: fixes A :: "'a set" shows "A \<inter> - A = {}"
slpa_callstack_length: fixes cs :: "'edge list"   and as :: "'edge list"   and cfsx :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cfs' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "same_level_path_aux cs as"   and "length cs = length cfsx" obtains cfx :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfsx' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "transfers (kinds as) (cfsx @ cf # cfs) = cfsx' @ cfx # cfs" and "transfers (kinds as) (cfsx @ cf # cfs') = cfsx' @ cfx # cfs'" and "length cfsx' = length (upd_cs cs as)"
Pure: fixes x :: "'d"   and gpv :: "('d, 'e, 'f) gpv"   and \<Gamma> :: "('e, 'f) \<I>" assumes "Pure x \<in> set_spmf (the_gpv gpv)" shows "x \<in> results_gpv \<Gamma> gpv"
assms_2: shows "dim_row A\<^sup>T = dim_vec c"
bot_entails_all: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and B :: "'a"   and N1 :: "'a set" assumes "consequence_relation Bot entails"   and "B \<in> Bot" shows "entails {B} N1"
raise_def: fixes f :: "'d \<Rightarrow> 'd" shows "raise f = Seq (Basic f) Throw"
list_ex_cong: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> bool"   and g :: "'a \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "list_ex f xs = list_ex g ys"
le_divide_eq_numeral1_2: fixes a :: "'a"   and b :: "'a"   and w :: "num" shows "(a \<le> b / - numeral w) = (b \<le> a * - numeral w)"
collapse_3: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and pr_operation' :: "pr_operation'" assumes "Network c s t"   and "is_SAT_PUSH' pr_operation'" shows "SAT_PUSH' (un_SAT_PUSH' pr_operation') = pr_operation'"
word_rotl_eq_word_rotr: fixes n :: "nat" shows "word_rotl n = word_rotr (LENGTH('b) - n mod LENGTH('b))"
finite_image_iff: fixes f :: "'q \<Rightarrow> 'r"   and A :: "'q set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
hom_def: fixes CC :: "('c, 'b, 'd) category_scheme"   and A :: "'c"   and B :: "'c" shows "Hom\<^bsub>CC\<^esub> A B = {f \<in> Ar\<^bsub>CC\<^esub>. Dom\<^bsub>CC\<^esub> f = A \<and> Cod\<^bsub>CC\<^esub> f = B}"
range_split_charn: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and p :: "'a \<Rightarrow> 'c decision option" shows "(f, g) \<nabla> p = (\<lambda>x. case p x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow x\<rfloor> \<Rightarrow> \<lfloor>allow (f x)\<rfloor> | \<lfloor>deny x\<rfloor> \<Rightarrow> \<lfloor>deny (g x)\<rfloor>)"
thd_tllist_of_llist: fixes b :: "'b"   and xs :: "'a llist" shows "thd (tllist_of_llist b xs) = lhd xs"
lappT: fixes s :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "s \<in> A\<^sup>\<infinity>"   and "t \<in> A\<^sup>\<infinity>" shows "s @@ t \<in> A\<^sup>\<infinity>"
n_restrictr: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> !y \<le> x"
61454: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - (- y \<sqinter> - x)) = - y \<sqinter> - x"
map_scale_spmf: fixes f :: "'b \<Rightarrow> 'a"   and r :: "real"   and p :: "'b spmf" shows "map_spmf f (scale_spmf r p) = scale_spmf r (map_spmf f p)"
fmlookup_dom_iff: fixes x :: "'d"   and m :: "'d \<rightharpoonup>\<^sub>f 'e" shows "(x |\<in>| fmdom m) = (\<exists>a. fmlookup m x = Some a)"
OneAncestor_HA: fixes HA :: "('a, 'b, 'c) hierauto" shows "OneAncestor (SAs HA) (CompFun HA)"
linepath_trivial: fixes a :: "'b"   and x :: "real" shows "linepath a a x = a"
assms_1: shows "a' \<le> nat \<lfloor>a\<rfloor>"
simps: fixes m :: "nat"   and n :: "nat" shows "mv_box m n = [Dec m 3, Inc n, Goto 0]"
cond_action_oks_conv_set: fixes s :: "('d, 'e) locks \<times> (('e \<Rightarrow> ('x1 \<times> 'd \<Rightarrow>f nat) option) \<times> 'f) \<times> ('e \<Rightarrow> 'g wait_set_status option) \<times> 'e set"   and t :: "'e"   and cts :: "'e conditional_action list" shows "r1.cond_action_oks s t cts = (\<forall>ct\<in>set cts. r1.cond_action_ok s t ct)"
Suffixes_UNION: fixes f :: "'b \<Rightarrow> 'a list set"   and A :: "'b set" shows "Suffixes (\<Union> (f ` A)) = (\<Union>x\<in>A. Suffixes (f x))"
EH\<^sub>N_def: fixes p :: "real"   and n :: "nat" shows "random_skip_list.EH\<^sub>N p n = measure_pmf.expectation (random_skip_list.H\<^sub>N p n) real"
fail: fixes C :: "'d \<Rightarrow> bool"   and r :: "'d"   and B :: "'d \<Rightarrow> 'd option" assumes "C r"   and "B r = None" shows "(Some r, None) \<in> option_while' C B"
eta: shows "t \<rightarrow>\<^sub>\<eta>\<^sup>* u"
encode_problem_parallel_correct_xi: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and ops :: "'b strips_operator list"   and op :: "'b strips_operator" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "ops \<in> set (\<Phi>\<inverse> \<Pi> \<A> t)"   and "op \<in> set ops" shows "op \<in> set (\<Pi>\<^sub>\<O>)"
chain_total: fixes C :: "'a set"   and x :: "'a"   and y :: "'a" assumes "chain_on (\<sqsubset>) C A"   and "x \<in> C"   and "y \<in> C" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
\<gamma>: shows "\<gamma> piecewise_C1_differentiable_on {a..b}"
matches_simps_1: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c"   and m' :: "'b match_expr" assumes "matches \<gamma> m a p" shows "matches \<gamma> (MatchAnd m m') a p = matches \<gamma> m' a p"
iT_Minus_image_conv: fixes k :: "nat"   and I :: "nat set" shows "k \<ominus> I = (-) k ` I \<down>\<le> k"
image_add_ball: fixes b :: "'a"   and a :: "'a"   and r :: "real" shows "(+) b ` ball a r = ball (a + b) r"
extremum_uniqueI: fixes a :: "nat" assumes "order.greater_eq 0 a" shows "a = 0"
c: shows "abc_steps_l (0, lm) ap stp = (as', am')"
inf_left_commute: fixes x :: "'d upred"   and y :: "'d upred"   and z :: "'d upred" shows "(x \<and> y \<and> z) = (y \<and> x \<and> z)"
comp_assoc: fixes h :: "'a"   and g :: "'a"   and f :: "'a" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
pref_prolong: fixes w :: "'b list"   and z :: "'b list"   and r :: "'b list"   and s :: "'b list" assumes "w \<le>p z \<cdot> r"   and "r \<le>p s" shows "w \<le>p z \<cdot> s"
dist_self: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.metric_space dist uniformity open" shows "dist x x = 0"
3: fixes x :: "'a"   and y :: "'a" assumes "fst (rep_nat_term x) = fst (rep_nat_term y)"   and "\<not> snd (rep_nat_term x) < snd (rep_nat_term y)"   and "x < y" shows "False"
run_owner_iii_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_owner_iii n s))) \<notin> used s"
restrict_apply: fixes f :: "'g \<Rightarrow> 'f"   and A :: "'g set"   and x :: "'g" shows "restrict f A x = (if x \<in> A then f x else undefined)"
assm_4: fixes i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "i \<in> I"   and "(x, K) \<in> pfn i" shows "\<exists>a b. K = cbox a b"
no_call_in_tail: fixes x :: "'a"   and p :: "'a list" shows "no_call_in_path x (tl p) = (no_call_in_path x p \<or> one_call_in_path x p \<and> hd p = x)"
connected_spanning_imp_connected: fixes H :: "('a, 'b) pre_digraph" assumes "spanning H G"   and "Digraph_Component.connected H" shows "Digraph_Component.connected G"
distinct_9: fixes x21 :: "char list"   and x22 :: "aexp"   and x31 :: "com"   and x32 :: "com" shows "x21 ::= x22 \<noteq> x31;; x32"
Fishb_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and A :: "'b set" shows "Fishb R B A = ((\<forall>a\<in>A. \<forall>b\<in>B - A. R b a) \<and> (\<forall>a\<in>A - B. \<forall>b\<in>B. R b a))"
list_merge: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "list (merge l r) = splice (list l) (list r)"
m_def: shows "m = drop_perp (Rep_hyp2 c) l"
atLeastatMost_empty: fixes b :: "'a"   and a :: "'a" assumes "b < a" shows "{a..b} = {}"
assms_1: fixes a :: "'a state" shows "pre_post (mk a) = a"
swapAbs_preserves_wls: fixes xs :: "'varSort"   and x :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "wlsAbs (xs, x) A" shows "wlsAbs (xs, x) (A $[z1 \<and> z2]_zs)"
WellOrderedStrategies_axioms_def: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set" shows "WellOrderedStrategies_axioms G S p good r \<equiv> (S \<subseteq> V\<^bsub>G\<^esub> \<and> well_order_on {\<sigma>. \<exists>v\<in>S. \<sigma> \<in> good v} r) \<and> (\<forall>v. v \<in> S \<longrightarrow> (\<exists>\<sigma>. \<sigma> \<in> good v)) \<and> (\<forall>v \<sigma>. \<sigma> \<in> good v \<longrightarrow> ParityGame.strategy G p \<sigma>) \<and> (\<forall>v w \<sigma>. v \<in> S \<longrightarrow> v \<rightarrow>\<^bsub>G\<^esub> w \<longrightarrow> (v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<longrightarrow> \<sigma> v = w) \<longrightarrow> \<sigma> \<in> good v \<longrightarrow> \<sigma> \<in> good w)"
hyp': shows "(l = PAodv-:4 \<or> l = PAodv-:5 \<or> l = PRreq-:0 \<or> l = PRreq-:2) \<and> sip (\<sigma> i) \<noteq> oip (\<sigma> i)"
take_conc: fixes xs :: "int list" shows "\<exists>ys. xs = Challenge1A.take xs @ ys"
circ_L: shows "L\<^sup>\<circ> = L \<squnion> (1::'a)"
select_convs_1: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "red \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = red"
PiE_eq_iff: fixes I :: "'a set"   and F :: "'a \<Rightarrow> 'b set"   and F' :: "'a \<Rightarrow> 'b set" shows "(Pi\<^sub>E I F = Pi\<^sub>E I F') = ((\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
cod_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.cod f = (if B.VVV.codomains f \<noteq> {} then SOME b. b \<in> B.VVV.codomains f else B.VVV.null)"
assms_2: shows "R\<^sup>+\<^sup>+ x y"
cases: fixes x :: "'b t_type" obtains uu_ :: "'b" and uv_ :: "'b bintree" where "x = \<lparr>t_type.folding = True, item = uu_, subtrees = [uv_]\<rparr>"   | uw_ :: "'b" and ux_ :: "'b bintree" and uy_ :: "'b bintree list" where "x = \<lparr>t_type.folding = True, item = uw_, subtrees = ux_ # Leaf # uy_\<rparr>"   | va :: "'b" and vb :: "'b bintree list" and vc :: "unit" where "x = \<lparr>t_type.folding = False, item = va__, subtrees = vb__, \<dots> = vc__\<rparr>"   | v :: "bool" and va :: "'b" and vc :: "unit" where "x = \<lparr>t_type.folding = v__, item = vaa__, subtrees = [], \<dots> = vca__\<rparr>"   | va :: "'b" and vd :: "'b bintree" and vf :: "'b bintree" and vg :: "'b bintree list" and vc :: "unit" where "x = \<lparr>t_type.folding = False, item = vab__, subtrees = vd__ # vf # vg__, \<dots> = vcb__\<rparr>"   | v :: "bool" and va :: "'b" and vd :: "'b bintree" and vb :: "'b" and ve :: "'b bintree" and vh :: "'b bintree" and vg :: "'b bintree list" and vc :: "unit" where "x = \<lparr>t_type.folding = v, item = va, subtrees = vd # Branch vb ve vh # vg, \<dots> = vc\<rparr>"
that: fixes B :: "'b set" assumes "finite B"   and "B \<subseteq> S"   and "independent B"   and "span B = S"   and "card B = dim S" shows "thesis"
listsetD_nth: fixes xs :: "'a list"   and Xs :: "'a set list" assumes "xs \<in> listset Xs" shows "\<forall>n<length xs. xs ! n \<in> Xs ! n"
assms_2: shows "X \<in>\<^sub>\<circ> Vset i"
assms_2: shows "i \<notin> net_tree_ips p"
WTVal: fixes v :: "val"   and T :: "ty"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option" assumes "typeof v = \<lfloor>T\<rfloor>" shows "P,E \<turnstile> Val v :: T"
le_diff_swap: fixes i :: "nat"   and k :: "nat"   and j :: "nat" assumes "i \<le> k"   and "j \<le> k" shows "(k - j \<le> i) = (k - i \<le> j)"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Num of Row \<Rightarrow> f1 | Col \<Rightarrow> f2 | Num \<Rightarrow> f3) = f3"
HStartBallot_HInv4b_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "Inv2a_innermost s p (dblock s p)"   and "HInv4b s p"   and "HInv4a s p" shows "HInv4b s' p"
1: shows "dom\<^bsub>C\<^esub> (id\<^bsub>C\<^esub> Y) = Y"
assms_4: shows "\<mu> \<star> e = \<mu>' \<star> e"
is_unit_content_iff: fixes p :: "'b poly" shows "is_unit (content p) = (content p = (1::'b))"
plus_minus_sub: fixes a :: "'a" assumes "a \<in> carrier R" shows "Cring_Poly.compose R (X_poly_plus R a) (X_poly_minus R a) = X_poly R"
set_fold_remove1: fixes xs :: "'b list"   and ys :: "'b list" shows "set (fold remove1 xs ys) \<subseteq> set ys"
integral_diff: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x - g x = integral\<^sup>L M f - integral\<^sup>L M g"
Cons: shows "\<forall>w1 w2. w \<in> w1 \<otimes> w2 \<longrightarrow> rev w \<in> rev w1 \<otimes> rev w2"
notMTnMT: fixes a :: "'b"   and p :: "'b list"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "a \<in> set p"   and "none_MT_rules C p" shows "dom (C a) \<noteq> {}"
restrict_af: fixes \<phi> :: "'b ltln"   and w :: "'b set list" shows "restricted_subformulas (af \<phi> w) = restricted_subformulas \<phi>"
append_same_eq: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" shows "(ys \<cdot> xs = zs \<cdot> xs) = (ys = zs)"
frequently_cofinite: fixes P :: "'b \<Rightarrow> bool" shows "Inf_many P = infinite {x. P x}"
gen_dfs_post_imp_below_U: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V0 :: "'a set"   and u0 :: "'a"   and V :: "'a set" assumes "gen_dfs_pre E U S V0 u0"   and "gen_dfs_post E U S V0 u0 V False" shows "V \<subseteq> U"
liveSimple': fixes n :: "'node"   and g :: "'g"   and val :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "val \<in> uses g n" shows "liveVal' g [val]"
inserts_tries_def: fixes key :: "'c \<Rightarrow> 'd list" shows "inserts_tries key = fold (\<lambda>v. insert_tries (key v) v)"
ex_sphere_params: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x * x + y * y + z * z = 1" shows "\<exists>\<alpha> \<beta>. x = cos \<alpha> * cos \<beta> \<and> y = cos \<alpha> * sin \<beta> \<and> z = sin \<alpha> \<and> - pi / 2 \<le> \<alpha> \<and> \<alpha> \<le> pi / 2 \<and> - pi \<le> \<beta> \<and> \<beta> < pi"
select_trap_result2: fixes s :: "'b sparc_state" assumes "\<not> (reset_trap_val s = False \<and> get_ET (cpu_reg_val PSR s) = 0)" shows "snd (select_trap () s) = False"
nat_mirror_diff: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and r :: "nat" assumes "k \<le> n"   and "n \<le> l + r" shows "nat_mirror (n - k) l r = nat_mirror n l r + k"
bounded_clinear_scaleC_left: fixes x :: "'a" shows "bounded_clinear (\<lambda>c. c *\<^sub>C x)"
finite_Un: fixes F :: "'e set"   and G :: "'e set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
exec_gpv_parametric': fixes S :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and CALL :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun CALL (rel_spmf (rel_prod R S)))) (rel_fun (rel_gpv'' A CALL R) (rel_fun S (rel_spmf (rel_prod A S)))) exec_gpv exec_gpv"
fact_dvd_higher_pderiv: fixes n :: "nat"   and p :: "int poly" shows "[:fact n:] dvd (pderiv ^^ n) p"
equiv_imp_le: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans t u"
Read_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "Read ch mm rs p \<equiv> \<exists>l. ReadInner ch mm rs p l"