hcju/isabelleps huggingface dataset:
technical: fixes w :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and w\<^sub>r :: "'a"   and \<theta>\<^sub>r :: "'a" assumes "ide w"   and "\<guillemotleft>\<theta> : (s\<^sub>0 \<star> p\<^sub>0) \<star> w \<Rightarrow> u\<guillemotright>"   and "w\<^sub>r = p\<^sub>1 \<star> w"   and "\<theta>\<^sub>r = (s \<star> \<theta>) \<cdot> (s \<star> \<a>\<^sup>-\<^sup>1[s\<^sub>0, p\<^sub>0, w]) \<cdot> \<a>[s, s\<^sub>0, p\<^sub>0 \<star> w] \<cdot> (\<sigma> \<star> p\<^sub>0 \<star> w) \<cdot> \<a>[s\<^sub>1, p\<^sub>0, w] \<cdot> (r\<^sub>0s\<^sub>1.\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[r\<^sub>0, p\<^sub>1, w]" shows "(r \<star> \<theta>\<^sub>r) \<cdot> \<a>[r, r\<^sub>0, w\<^sub>r] \<cdot> (\<rho> \<star> w\<^sub>r) = \<a>[r, s, u] \<cdot> (((r \<star> s) \<star> \<theta>) \<cdot> \<a>[r \<star> s, s\<^sub>0 \<star> p\<^sub>0, w] \<cdot> (tab \<star> w)) \<cdot> \<a>\<^sup>-\<^sup>1[r\<^sub>1, p\<^sub>1, w]"
sorted_less_set_unique: fixes xs :: "'a list"   and ys :: "'a list" assumes "Sorted_Less.sorted xs"   and "Sorted_Less.sorted ys"   and "set xs = set ys" shows "xs = ys"
valid_graph_repr_def: fixes l :: "('c \<times> 'c) list" shows "valid_graph_repr l = (\<forall>(u, v)\<in>set l. u \<noteq> v)"
share_and_reconstruct: fixes x :: "bool"   and y :: "bool" shows "share x \<bind> (\<lambda>(a1, a2). share y \<bind> (\<lambda>(b1, b2). and_protocol (a1, b1) (a2, b2) \<bind> (\<lambda>(a, b). reconstruct (a, b)))) = and_evaluate x y"
assms_1: shows "ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) c' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := c'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>"
strict_iff_order: fixes b :: "'t"   and a :: "'t" shows "(b \<prec>\<^sub>t a) = (b \<preceq>\<^sub>t a \<and> a \<noteq> b)"
eval_ww_def: shows "eval_ww \<equiv> wrap\<cdot>eval_work"
inext_nth_gr_Min_conv_infinite: fixes I :: "nat set"   and n :: "nat" assumes "infinite I" shows "(iMin I < I \<rightarrow> n) = (0 < n)"
wfP_subterm_T: shows "wfP (\<lambda>x y. x \<in> set3_F\<^sub>m (the_T\<^sub>m y))"
n_preserve2: fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> !y \<cdot> t x = o"
dom: shows "\<forall>ps n. I (ps, n) \<longrightarrow> vars b \<subseteq> dom ps"
distinct_cols_submatrix_UNIV: fixes A :: "'b mat"   and I :: "nat set" assumes "distinct (cols (submatrix A I UNIV))" shows "distinct (cols A)"
antisym_conv1: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "\<not> x \<subset># y" shows "(x \<subseteq># y) = (x = y)"
map_closed: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and a :: "'a" assumes "map \<alpha> S T"   and "a \<in> S" shows "\<alpha> a \<in> T"
seqI: fixes f :: "'s"   and g :: "'s" assumes "arr f"   and "arr g"   and "local.dom g = cod f" shows "seq g f"
safetyI2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<And>\<sigma>. (\<And>i. \<exists>\<beta>. P (stake i \<sigma> @- \<beta>)) \<Longrightarrow> P \<sigma>" shows "safety P"
wt_Ch_L: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and s :: "'state" shows "wt (Ch ch c1 c2) s 0 = cval ch s"
invmode_def: fixes m :: "'a member_scheme"   and e :: "expr" shows "invmode m e = (if is_static m then Static else if e = Super then SuperM else IntVir)"
facet_of_def: fixes F :: "'a set"   and S :: "'a set" shows "(F facet_of S) = (F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1)"
bisimilar_def: fixes a :: "('e, 'f) Stream"   and b :: "('e, 'g) Stream" shows "(a \<approx> b) = (unstream\<cdot>a = unstream\<cdot>b \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
b_least2_aux1: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" shows "b_least2 f x y = y \<or> b_least2 f x y < y \<and> f x (b_least2 f x y) \<noteq> 0"
zero_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_1 one times plus zero"   and "0 < n" shows "power.power one times zero n = zero"
nargs_s_lt: shows "enat (num_args s) < arity_hd (head s)"
i_take_Suc_conv_app_nth: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "f \<Down> Suc n = f \<Down> n @ [f n]"
Mapping_keys_filterI: fixes m :: "('b, 'a) mapping"   and k :: "'b"   and v :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "Mapping.lookup m k = Some v"   and "f k v" shows "k \<in> Mapping.keys (Mapping.filter f m)"
1: fixes y' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F y' x f" shows "\<epsilon>x.is_coext y' f (\<phi> y' f) \<and> (\<forall>g'. \<epsilon>x.is_coext y' f g' \<longrightarrow> g' = \<phi> y' f)"
r0: fixes x :: "rat" assumes "x < r0" shows "f x = (0::'a)"
CA_Says_4_lemma: fixes i :: "nat"   and C :: "agent"   and NC2 :: "nat"   and NCA :: "nat"   and evs :: "event list" assumes "Crypt (priSK (CA i)) (Hash \<lbrace>Agent C, Nonce NC2, Nonce NCA\<rbrace>) \<in> parts (knows Spy evs)"   and "evs \<in> set_cr"   and "CA i \<notin> bad" shows "\<exists>Y. Says (CA i) C \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce NC2, Nonce NCA\<rbrace>, Y\<rbrace> \<in> set evs"
SpecAnno': fixes P :: "'d set"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and Q :: "'d set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('d, 'f, 'g) com" assumes "P \<subseteq> {s. \<exists>Z. s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> t \<in> Q) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> t \<in> A)}"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c Z (Q' Z),(A' Z)"   and "\<forall>Z. c Z = c undefined" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P specAnno P' c Q' A' Q,A"
sorted_wrt_keys_map_fst: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and l :: "('d \<times> 'e) list" shows "sorted_wrt (\<lambda>(k, uu_) (k', uu_). R k k') l = sorted_wrt R (map fst l)"
option_ord_Some1_iff: fixes x :: "'c"   and y :: "'c option" shows "option_ord (Some x) y = (y = Some x)"
Refl: fixes P :: "ccs" shows "P \<equiv>\<^sub>s P"
2: shows "e\<^sup>T * top * e = e\<^sup>T * e"
min_0R: fixes n :: "V" shows "min n 0 = 0"
unit_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (F.unit a) = F.map\<^sub>0 a"
SReal_iff: fixes x :: "real star" shows "(x \<in> \<real>) = (\<exists>y. x = hypreal_of_real y)"
ETP_sec_para_axioms: shows "ETP_sec_para I domain range F F\<^sub>i\<^sub>n\<^sub>v"
abs_eq: fixes x :: "real \<times> real" assumes "eq_onp (\<lambda>r. fst r \<le> snd r) x x" shows "right (Abs_real_int x) = snd x"
exhaust_disc: fixes pr_operation :: "pr_operation" obtains "is_PUSH pr_operation"   | "is_RELABEL pr_operation"
hs: assumes "A \<rightarrow>\<^sub>\<S> C" shows "A \<rightarrow>\<^sub>\<S> C + D"
Lunit_in_lunit: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "arr f"   and "t \<in> f" shows "\<^bold>\<l>\<^bold>[t\<^bold>] \<in> \<l>[f]"
assms: fixes B :: "fm" assumes "atom i \<sharp> B" shows "insert A H \<turnstile> B"
lim_imp_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and f0 :: "'c" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> f0) F" shows "Limsup F f = f0"
cswitch_consistency: shows "\<forall>u s t n. vpeq u s t \<longrightarrow> vpeq u (cswitch n s) (cswitch n t)"
lsc_hull_iff_greatest: fixes g :: "'b \<Rightarrow> ereal"   and f :: "'b \<Rightarrow> ereal" shows "(g = lsc_hull f) = (lsc g \<and> (\<forall>x. g x \<le> f x) \<and> (\<forall>h. lsc h \<and> (\<forall>x. h x \<le> f x) \<longrightarrow> (\<forall>x. h x \<le> g x)))"
SkipD: fixes G :: "prog"   and s :: "state"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>Skip\<rightarrow> s'" shows "s' = s"
map_default_empty: fixes b :: "'b" shows "map_default b Map.empty = (\<lambda>a. b)"
satisfiableEquivalent: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list" assumes "equivalentFormulae formula1 formula2" shows "satisfiable formula1 = satisfiable formula2"
vdisjnt_singleton2: fixes A :: "V"   and a :: "V" shows "(A \<inter>\<^sub>\<circ> set {a} = []\<^sub>\<circ>) = (a \<notin>\<^sub>\<circ> A)"
transUpdate_def: fixes ec :: "'rep"   and ec' :: "'rep"   and at :: "'mt" shows "transUpdate ec ec' at \<equiv> update tOps (ec, simObs a ec') ec' at"
inj_on_inv_vreal_of_real: shows "inj_on (inv vreal_of_real) (elts \<real>\<^sub>\<circ>)"
compositionality_R: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>1 :: "'b V_rec"   and \<V>2 :: "'b V_rec" assumes "Compositionality ES1 ES2 \<V> \<V>1 \<V>2"   and "R \<V>1 Tr\<^bsub>ES1\<^esub>"   and "R \<V>2 Tr\<^bsub>ES2\<^esub>" shows "R \<V> Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"
FrameStatEqTrans: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and G :: "'d frame"   and H :: "'d frame" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.FrameStatEq SImp F G"   and "assertionAux.FrameStatEq SImp G H" shows "assertionAux.FrameStatEq SImp F H"
1: shows "f (local.lfp f) \<le> local.lfp f"
zmset_of_subseteq_iff: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(zmset_of M \<subseteq>#\<^sub>z zmset_of N) = (M \<subseteq># N)"
ZObis_pres_discr_R: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "discr c"   and "c \<approx>01 d" shows "discr d"
hom_1_iff: fixes x :: "'a mod_ring" shows "(to_int_mod_ring x = 1) = (x = 1)"
before_dist_snd: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "distinct vs" shows "fst (splitAt ram1 (snd (splitAt ram2 vs))) = snd (splitAt ram2 vs)"
has_integral_box: fixes l :: "'a"   and u :: "'a" shows "((\<lambda>x. 1) has_integral content (box l u)) (box l u)"
assms_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "R\<^sup>+\<^sup>+ x y"   and "R y z"   and "P x"   and "P z" shows "P y"
absolutely_integrable_mult_Fejer_kernel_reflected_part1: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and n :: "nat"   and t :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "d \<le> pi" shows "(\<lambda>x. Fejer_kernel n x * f (t + x)) absolutely_integrable_on {0..d}"
arev_in_arcs: fixes a :: "'b" assumes "a \<in> arcs G" shows "rev_G a \<in> arcs G"
assms: shows "<\<nu>x>a{b}.P \<longmapsto> c<y> \<prec> P'"
semigroup_const: shows "semigroup (\<lambda>x y. x)"
card_distinct: fixes xs :: "'c list" assumes "card (set xs) = length xs" shows "distinct xs"
TE_Tag: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and f :: "'a"   and al :: "univ list" assumes "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP"   and "wtFsym f"   and "list_all2 eintT (arOf f) al" shows "eintF (Tag (resOf f)) [eintF (Oldf f) al] = eintF (Oldf f) al"
integral_iff: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "(a \<otimes> b = \<zero>) = (a = \<zero> \<or> b = \<zero>)"
vimage_algebra_sigma: fixes X :: "'c set set"   and \<Omega>' :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and \<Omega> :: "'d set" assumes "X \<subseteq> Pow \<Omega>'"   and "f \<in> \<Omega> \<rightarrow> \<Omega>'" shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f -` A \<inter> \<Omega> |A. A \<in> X}"
map\<^sub>0_def: fixes a :: "'b" shows "F.map\<^sub>0 a \<equiv> src\<^sub>C (F a)"
coprimeE: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and p :: "'b"   and q :: "'b" assumes "class.comm_monoid_mult times one"   and "comm_monoid_mult.coprime times one p q" obtains "\<And>r. \<lbrakk>dvd.dvd times r p; dvd.dvd times r q\<rbrakk> \<Longrightarrow> dvd.dvd times r one"
abelian_group_def: fixes G :: "('a, 'b) ring_scheme" shows "abelian_group G \<equiv> abelian_monoid G \<and> abelian_group_axioms G"
x_lt_xs: shows "\<forall>v\<in>set xs. x < v"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "partially_ordered_set A less_eq" shows "reflexive A less_eq"
theI': fixes P :: "'e \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
s_top_induced: fixes G :: "('e, 'f) Group_scheme"   and L :: "'e set"   and H :: "'e set"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> L "   and "H \<subseteq> L"   and "K \<subseteq> L" shows "H \<diamondop>\<^bsub>\<natural>\<^bsub>G\<^esub>L\<^esub> K = H \<diamondop>\<^bsub>G\<^esub> K"
product_overlap: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "P z x"   and "O z y" shows "O z (x \<otimes> y)"
l10_2_uniqueness: fixes P1 :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p"   and P2 :: "'p" assumes "P1 P Reflect A B"   and "P2 P Reflect A B" shows "P1 = P2"
intuitionistic_def: fixes P :: "'a \<Rightarrow> bool" shows "intuitionistic P \<equiv> \<forall>h h'. P h \<and> h \<preceq> h' \<longrightarrow> P h'"
starfun_inverse_real_of_nat_eq: fixes N :: "nat star" assumes "N \<in> HNatInfinite" shows "(*f* (\<lambda>x. inverse (real x))) N = inverse (hypreal_of_hypnat N)"
trcl_empty_simp: fixes c :: "'f"   and c' :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set" shows "((c, [], c') \<in> trcl t) = (c = c')"
invc_baliL: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invc2 l"   and "invc r" shows "invc (baliL l a r)"
mstfe_no_repetition_2: fixes V :: "'d list set"   and M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V'' :: "('d \<times> 'e) list set"   and xs' :: "('d \<times> 'e) list" assumes "minimal_sequence_to_failure_extending V M1 M2 vs xs"   and "OFSM M1"   and "OFSM M2"   and "test_tools M2 M1 FAIL PM V \<Omega>"   and "V'' \<in> N (vs @ xs') M1 V"   and "prefix xs' xs" shows "\<not> Rep_Cov M2 M1 V'' vs xs'"
is_OK_returns_result_E: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog" assumes "h \<turnstile> ok f" obtains x :: "'o" where "h \<turnstile> f \<rightarrow>\<^sub>r x"
mat_of_rows_def: fixes n :: "nat"   and rs :: "'c vec list" shows "mat_of_rows n rs = mat (length rs) n (\<lambda>(i, j). rs ! i $ j)"
not_Cons_self2: fixes x :: "'b"   and xs :: "'b list" shows "Cons x xs \<noteq> xs"
wellformed: fixes b :: "'b set" assumes "b \<in># \<B>'" shows "b \<subseteq> \<V>'"
VSHeuristic_def: shows "VSHeuristic = opt_group \<circ> QE_dnf opt_group superPicker \<circ> VSLuckiestBlocks \<circ> opt_group"
ab_left_minus: fixes a :: "'ag" assumes "a \<in> U" shows "-\<^sub>o\<^sub>w a +\<^sub>o\<^sub>w a = 0\<^sub>o\<^sub>w"
is_oc_projection_def: fixes w :: "'a vec"   and S :: "'a vec set"   and v :: "'a vec" shows "is_oc_projection w S v = (w \<in> carrier_vec n \<and> v - w \<in> local.span S \<and> (\<forall>u. u \<in> S \<longrightarrow> w \<bullet> u = (0::'a)))"
lm036: fixes P :: "('d \<times> 'e) set"   and X :: "'d set" shows "P || X = P outside (Domain P - X)"
assms_2: shows "F \<in> \<lbrakk>id\<rbrakk>\<^sub>H \<rightarrow> \<lbrakk>H\<rbrakk>\<^sub>H"
l_neg: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier (func_space S)" shows "\<ominus>\<^bsub>func_space S\<^esub> x \<oplus>\<^bsub>func_space S\<^esub> x = \<zero>\<^bsub>func_space S\<^esub>"
homeomorphic_compact: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and t :: "'c set" assumes "compact s"   and "continuous_on s f"   and "f ` s = t"   and "inj_on f s" shows "s homeomorphic t"
sym: fixes x :: "'a"   and y :: "'a" assumes "x .= y"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "y .= x"
step: shows "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', mis)"
share_eq: shows "share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L"
LESPLIT_4: shows "(ch, w2, c2') \<in> trcl (ntr fg)"
cong_sym: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo c b a"
empty_subsetI: fixes A :: "'q set" shows "{} \<subseteq> A"
iview_partitioned_def: shows "iview_partitioned \<equiv> \<forall>execs ms mt n u. iequivalent_states ms mt u \<longrightarrow> B2 (run n ms (ipurge_l execs u)) (run n mt (ipurge_r execs u)) (\<lambda>rs rt. vpeq u rs rt \<and> current rs = current rt)"
nn_integral_eq_integral: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal (integral\<^sup>L M f)"
abs_eq: fixes x :: "nat \<Rightarrow> 'c" shows "of_seq x = stream_of_llist (inf_llist x)"
map\<^sub>0_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (G.map\<^sub>0 a) = G.map\<^sub>0 a"
polypoly'_poly: fixes p :: "'b poly"   and n0 :: "nat"   and x :: "'b"   and bs :: "'b list" assumes "isnpolyh p n0" shows "Ipoly (x # bs) p = poly (polypoly' bs p) x"
invariant_def: fixes A :: "'c set" shows "invariant A \<equiv> {F. Init F \<subseteq> A} \<inter> stable A"
repeat_mset_eq_empty_iff: fixes n :: "nat"   and A :: "'b multiset" shows "(repeat_mset n A = {#}) = (n = 0 \<or> A = {#})"
distinct_replacefacesAt1_norm: fixes i :: "nat"   and Fss :: "face list list"   and "is" :: "nat list"   and newFs :: "face list"   and oldF :: "face" assumes "i < |Fss|"   and "i \<in> set is"   and "distinct is"   and "distinct (normFaces (Fss ! i))"   and "distinct (normFaces newFs)"   and "set (normFaces (Fss ! i)) \<inter> set (normFaces newFs) \<subseteq> {}" shows "distinct (normFaces (replacefacesAt is oldF newFs Fss ! i))"
less_bot_Bot_Value: fixes x :: "'b" shows "bot.Bot < bot.Value x"
zero_left: fixes b :: "'b" shows "cblinfun_apply 0 b = (0::'a)"
eval_fps_cos: fixes c :: "'a"   and z :: "'a" shows "eval_fps (fps_cos c) z = cos (c * z)"
simps: fixes i :: "nat" shows "f2.gso i = f2 ! i + M.sumlist (map (\<lambda>j. - f2.\<mu> i j \<cdot>\<^sub>v f2.gso j) [0..<i])"
cases: fixes a1 :: "cmd \<times> (char list \<Rightarrow> val option)"   and a2 :: "cmd \<times> (char list \<Rightarrow> val option)" assumes "red a1 a2" obtains V :: "char list" and e :: "expr" and s :: "char list \<Rightarrow> val option" where "a1 = (V:=e, s__)" and "a2 = (Skip, s__(V := interpret e s__))"   | c\<^sub>1 :: "cmd" and s :: "char list \<Rightarrow> val option" and c\<^sub>1' :: "cmd" and s' :: "char list \<Rightarrow> val option" and c\<^sub>2 :: "cmd" where "a1 = (c\<^sub>1__;; c\<^sub>2__, sa__)" and "a2 = (c\<^sub>1';; c\<^sub>2__, s')" and "\<langle>c\<^sub>1__,sa__\<rangle> \<rightarrow> \<langle>c\<^sub>1',s'\<rangle>"   | c\<^sub>2 :: "cmd" and s :: "char list \<Rightarrow> val option" where "a1 = (Skip;; c\<^sub>2'__, sb__)" and "a2 = (c\<^sub>2'__, sb__)"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c\<^sub>1 :: "cmd" and c\<^sub>2 :: "cmd" where "a1 = (if (b__) c\<^sub>1''__ else c\<^sub>2''__, sc__)" and "a2 = (c\<^sub>1''__, sc__)" and "interpret b__ sc__ = Some true"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c\<^sub>1 :: "cmd" and c\<^sub>2 :: "cmd" where "a1 = (if (ba__) c\<^sub>1 else c\<^sub>2, sd__)" and "a2 = (c\<^sub>2, sd__)" and "interpret ba__ sd__ = Some false"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c :: "cmd" where "a1 = (while (bb__) c__, se__)" and "a2 = (c__;; while (bb__) c__, se__)" and "interpret bb__ se__ = Some true"   | b :: "expr" and s :: "char list \<Rightarrow> val option" and c :: "cmd" where "a1 = (while (b) c, s)" and "a2 = (Skip, s)" and "interpret b s = Some false"
or_wf_dfa: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and N :: "nat bdd list \<times> bool list" assumes "wf_dfa M n"   and "wf_dfa N n" shows "wf_dfa (or_dfa M N) n"
q_eq2: shows "int q = 2 * ((int q - 1) div 2) + 1"
bot_or_not_bot': fixes n :: "nat"   and x :: "Value"   and y :: "CValue" assumes "x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> y" shows "(x = \<bottom>) = (y = \<bottom>)"
induced_modulus_altdef1: fixes d :: "nat" shows "induced_modulus d = (d dvd n \<and> (\<forall>a b. coprime a n \<and> coprime b n \<and> [a = b] (mod d) \<longrightarrow> \<chi> a = \<chi> b))"
simps_1: fixes ParamDefs :: "'h \<Rightarrow> 'g list"   and n :: "'h" shows "lift_ParamDefs ParamDefs (Node n) = ParamDefs n"
plus_matrix_def: fixes A :: "('c, 'd, 'e) matrix"   and B :: "('c, 'd, 'e) matrix" shows "A + B \<equiv> matrix_plus A B"
g_def: shows "g = Polynomial.smult (lead_coeff v) w"
isMaxim_def: fixes A :: "'c set"   and b :: "'c" shows "t.isMaxim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (a, b) \<in> t)"
l_set_disconnected_nodes_get_disconnected_nodes_wf_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_disconnected_nodes_get_disconnected_nodes_wf type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs \<equiv> l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes \<and> l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs \<and> l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms heap_is_wellformed get_disconnected_nodes set_disconnected_nodes"
T9: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "|~ \<box>[A]_v \<longrightarrow> _actrans A v"