hcju/isabelleps huggingface dataset:
mod_div_trivial: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "divide (modulo a b) b = zero"
prim_impl_correct_1: fixes g :: "(('a \<times> ('a \<times> nat) list) \<times> color) tree"   and r :: "'a"   and \<pi>i :: "(('a \<times> 'a) \<times> color) tree" assumes "G.invar g"   and "prim_impl g r = \<pi>i" shows "P.invar_MST \<pi>i"
intro: fixes A :: "V"   and f :: "V" assumes "nop_onto A (1\<^sub>\<nat>) f" shows "unop_onto A f"
extract_res_def: fixes cyc :: "'a blue_witness" shows "extract_res cyc \<equiv> case cyc of NO_CYC \<Rightarrow> None | REACH v x \<Rightarrow> Map.empty x | CIRC pr pl \<Rightarrow> Some (pr, pl)"
LB_def: fixes A :: "int set"   and n :: "int" shows "LB A n = (\<forall>a\<in>A. n \<le> a)"
lower_uminus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lower (- p) v = - lower p v"
assms_3: shows "src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>)"
A: shows "\<forall>s. s \<in> wa_invar WAA \<and> s \<notin> wa_cond WAA \<longrightarrow> P s"
sel_1: fixes x :: "quat" shows "quat.Re (inverse x) = quat.Re x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2)"
simps_2: fixes \<A> :: "'a ltl set" shows "\<A> \<Turnstile>\<^sub>P false = False"
empty_is_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "({} = f ` A) = (A = {})"
rel_fun_mono': fixes Y :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<longrightarrow> B x y" shows "rel_fun X A f g \<longrightarrow> rel_fun Y B f g"
clock_monotone_2: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and es :: "exp0 list"   and r2 :: "'b state \<times> (v list, v) result" assumes "evaluate_list ck env s es r2" shows "\<forall>s' r'. r2 = (s', r') \<and> ck = True \<longrightarrow> clock s' \<le> clock s"
assms: shows "\<S>\<^sub>1 \<hookrightarrow> \<S>\<^sub>2"
scc_ofI_reachable': fixes v :: "'a"   and u :: "'a" assumes "v \<rightarrow>\<^sup>* u" shows "u \<in> scc_of v"
P_def: shows "P = (\<lambda>h qf. finite (fst qf) \<and> fst qf \<subseteq> F \<and> h = (\<Sum>f\<in>fst qf. snd qf f * f) \<and> (\<forall>f\<in>fst qf. poly_deg (snd qf f * f) = poly_deg h) \<and> (\<forall>f. f \<notin> fst qf \<longrightarrow> snd qf f = 0))"
alpha_tPred: fixes \<phi>1 :: "'f"   and \<phi>2 :: "'f" shows "(tPred \<phi>1 =\<^sub>\<alpha> tPred \<phi>2) = (\<phi>1 = \<phi>2)"
antisym: fixes b :: "'e list"   and a :: "'e list" assumes "prefix b a"   and "prefix a b" shows "a = b"
nextl_state: fixes Q :: "hf set"   and xs :: "'a list" shows "nextl Q xs \<subseteq> nfa.states M"
FAccRed: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<rangle> \<rightarrow> \<langle>e'\<bullet>F{D},s'\<rangle>"
dense_eq0_I: fixes x :: "'a" assumes "\<And>e. (0::'a) < e \<Longrightarrow> \<bar>x\<bar> \<le> e" shows "x = (0::'a)"
case_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Row of Row \<Rightarrow> f1 | Col \<Rightarrow> f2 | Num \<Rightarrow> f3) = f1"
accept\<^sub>G\<^sub>R_I: fixes \<phi> :: "'a ltl" assumes "accept\<^sub>G\<^sub>R (\<A>.\<A> \<Sigma> \<phi>) w" obtains \<pi> :: "'a ltl \<Rightarrow> nat option" where "dom \<pi> \<subseteq> \<^bold>G \<phi>" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) < \<A>.max_rank_of \<Sigma> \<chi>" and "accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (M_fin \<pi>, UNIV) w" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (\<A>.Acc\<^sub>\<A> \<Sigma> \<pi> \<chi>) w"
arr_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "ide a"   and "ide b"   and "ide c" shows "arr \<a>[a, b, c]"
sum_SDG_slice2p_sum_SDG_slice2_eq: fixes n :: "'node SDG_node" shows "sum_SDG_slice2p n = (\<lambda>x. x \<in> sum_SDG_slice2 n)"
function_one_is_constant: fixes R :: "('a, 'b) ring_scheme"   and S :: "'c set" assumes "ring_functions R" shows "constant_function S \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>function_ring S R\<^esub>"
mult_right_cancel: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times a c = times b c) = (a = b)"
n_inter_num_choose: fixes b1 :: "'b set"   and b2 :: "'b set"   and n :: "nat" assumes "finite b1"   and "finite b2" shows "(b1 |\<inter>|\<^sub>n b2) = int (card (b1 \<inter> b2) choose n)"
represents_of_real: fixes p :: "int poly"   and x :: "real" shows "p represents of_real x = p represents x"
lm046: fixes a :: "('c \<times> 'd set) set" shows "pseudoAllocation a = {(fst pair, Y) |Y pair. Y \<in> finestpart (snd pair) \<and> pair \<in> a}"
infdist_pos_not_in_closed: fixes S :: "'b set"   and x :: "'b" assumes "closed S"   and "S \<noteq> {}"   and "x \<notin> S" shows "0 < infdist x S"
standard_decomp_def: fixes k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "standard_decomp k ps = (\<forall>(h, U)\<in>set (ps\<^sub>+). k \<le> poly_deg h \<and> (\<forall>d\<ge>k. d \<le> poly_deg h \<longrightarrow> (\<exists>(h', U')\<in>set ps. poly_deg h' = d \<and> card U \<le> card U')))"
r_neg1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> x \<oplus> (x \<oplus> y) = y"
path_entry_dom: fixes pa :: "nat list"   and i :: "nat"   and d :: "nat" assumes "path_entry (g_E G) pa i"   and "local.dom d i" shows "d \<in> set pa \<or> d = i"
Units_l_cancel: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "Group.monoid G"   and "x \<in> Units G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> y = x \<otimes>\<^bsub>G\<^esub> z) = (y = z)"
bool_taut_test_def: shows "bool_taut_test \<equiv> taut_test ifex_of"
tau_eqvt: fixes p :: "perm" shows "p \<bullet> \<tau> = \<tau>"
geq_imp_mix_geq_left: fixes x :: "'a pmf"   and y :: "'a pmf"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and outcomes :: "'a set"   and \<alpha> :: "real" assumes "x \<succeq>[\<R>] y"   and "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> y x \<succeq>[\<R>] y"
subcls1_wfD: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C \<prec>\<^sup>1 D"   and "wf_prog wf_md P" shows "D \<noteq> C \<and> (D, C) \<notin> (subcls1 P)\<^sup>+"
prv_tms_cong: fixes t1 :: "'trm"   and t1' :: "'trm"   and t2 :: "'trm"   and t2' :: "'trm" assumes "t1 \<in> atrm"   and "t1' \<in> atrm"   and "t2 \<in> atrm"   and "t2' \<in> atrm"   and "prv (eql t1 t1')"   and "prv (eql t2 t2')" shows "prv (eql (tms t1 t2) (tms t1' t2'))"
assoc'_in_hom_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<guillemotleft>\<a>\<^sup>-\<^sup>1[f, g, h] : local.dom f \<star> local.dom g \<star> local.dom h \<rightarrow> (cod f \<star> cod g) \<star> cod h\<guillemotright>"
RestrictedP_imp_Subset: fixes s :: "tm"   and k :: "tm"   and s' :: "tm" shows "{RestrictedP s k s'} \<turnstile> s' SUBS s"
add_0_right: fixes a :: "'c" shows "a \<squnion> bot = a"
coboundedI1: fixes a :: "'d"   and c :: "'d"   and b :: "'d" assumes "a \<le> c" shows "inf a b \<le> c"
f_def: shows "is_det_state_cover_ass M f \<and> V = f ` d_reachable M (initial M)"
assms_11: shows "bijective (a * top) \<and> bijective (a[[top]])"
has_exponentials: fixes b :: "'a"   and c :: "'a" assumes "ide b"   and "ide c" shows "\<exists>x e. ide x \<and> \<guillemotleft>e : local.prod x b \<rightarrow> c\<guillemotright> \<and> (\<forall>a g. ide a \<and> \<guillemotleft>g : local.prod a b \<rightarrow> c\<guillemotright> \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> x\<guillemotright> \<and> g = C e (local.prod f b)))"
Prim2_imp_R2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> Prim2" shows "f \<in> \<R>\<^sup>2"
hyps_4: shows "Cres = map (\<lambda>(F, Tm). ((F, C), Tm)) fs @ Dres"
inject_1: fixes a :: "'e"   and a' :: "'e" shows "(Inl a = Inl a') = (a = a')"
Call_par_red: fixes s :: "sterm"   and l :: "Label"   and u :: "sterm"   and z :: "sterm" assumes "Call s l u => z" obtains t :: "sterm" and u' :: "sterm" where "s => t" and "u => u'__" and "z = Call t l u'__"   | f :: "Label \<Rightarrow> sterm option" and f' :: "Label \<Rightarrow> sterm option" and T :: "Sigma.type" and u' :: "sterm" where "Obj f T = s" and "Obj f T => Obj f' T" and "l \<in> dom f'" and "u => u'" and "z = (the (f' l)\<^bsup>[Obj f' T,u']\<^esup>)"
restrict_indets_times: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "restrict_indets (p * q) = restrict_indets p * restrict_indets q"
p1: fixes z :: "'a" shows "p1 z \<in> T"
assms_2: shows "ideal \<bb> R (+) (\<cdot>) \<zero> \<one>"
assms: fixes a :: "nat"   and b :: "(nat \<times> char list) list"   and c :: "call" shows "cont (\<lambda>x. f x a b c)"
birkhoff_sum_1_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "qmpt M T" shows "qmpt.birkhoff_sum T f 0 x = (0::'c)"
prim_vc_2: fixes t :: "'a"   and v :: "'a"   and g :: "'a"   and r :: "'a" assumes "prim_spanning_invariant t v g r"   and "v * - v\<^sup>T \<sqinter> g \<noteq> bot" shows "prim_spanning_invariant (t \<squnion> minarc (v * - v\<^sup>T \<sqinter> g)) (v \<squnion> minarc (v * - v\<^sup>T \<sqinter> g)\<^sup>T * top) g r \<and> card {x. regular x \<and> x \<le> component g r \<and> x \<le> - (v \<squnion> minarc (v * - v\<^sup>T \<sqinter> g)\<^sup>T * top)\<^sup>T} < card {x. regular x \<and> x \<le> component g r \<and> x \<le> - v\<^sup>T}"
not_coprimeI: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c dvd a"   and "c dvd b"   and "\<not> is_unit c" shows "\<not> coprime a b"
dd2: fixes l :: "'a"   and e :: "real" assumes "a + l \<in> affine hull S"   and "e < dd l"   and "0 \<le> e"   and "l \<noteq> (0::'a)" shows "a + e *\<^sub>R l \<in> rel_interior S"
subtermeq_imp_funs_term_subset: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" assumes "s \<sqsubseteq> t" shows "funs_term s \<subseteq> funs_term t"
sequent_semantics_def: fixes \<A> :: "'b \<Rightarrow> bool"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "\<A> \<Turnstile> \<Gamma> \<Rightarrow> \<Delta> \<equiv> (\<forall>\<gamma>\<in>#\<Gamma>. \<A> \<Turnstile> \<gamma>) \<longrightarrow> (\<exists>\<delta>\<in>#\<Delta>. \<A> \<Turnstile> \<delta>)"
le_utrans_antisym: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_utrans x y"   and "le_utrans y x" shows "equiv_utrans x y"
qcr_dlist_def: fixes x :: "'d list"   and y :: "'d dlist" shows "qcr_dlist x y = (y = abs_dlist x)"
encode_goal_state_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "(\<Phi>\<^sub>G \<Pi>) t \<equiv> let vs = \<Pi>\<^sub>\<V>; G = \<Pi>\<^sub>G in \<^bold>\<And>map (\<lambda>v. encode_state_variable t (index vs v) (G v) \<^bold>\<or> \<bottom>) (filter (\<lambda>v. G v \<noteq> None) vs)"
add_new_coeff_l: fixes x :: "char list multiset \<times> int"   and p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset" shows "add_poly_p (add_mset x p, q, r) (p, q, add_mset x r)"
diff_empty: fixes M :: "'b multiset" shows "M - {#} = M \<and> {#} - M = {#}"
minus_monom: fixes a :: "'a"   and n :: "nat" shows "- monom a n = monom (- a) n"
cblinfun_image_id: fixes \<psi> :: "'a ccsubspace" shows "id_cblinfun *\<^sub>S \<psi> = \<psi>"
dictates2_def: fixes F :: "(indi \<Rightarrow> alt \<Rightarrow> real) \<Rightarrow> alt \<Rightarrow> real"   and i :: "indi"   and a :: "alt"   and b :: "alt" assumes "arrow F" shows "arrow.dictates2 F i a b \<equiv> arrow.dictates F i a b \<and> arrow.dictates F i b a"
UNIV_eq: shows "UNIV = Abs ` {0..<n}"
nn_integral_distr: fixes T :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and M' :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "T \<in> M \<rightarrow>\<^sub>M M'"   and "f \<in> borel_measurable (distr M M' T)" shows "integral\<^sup>N (distr M M' T) f = \<integral>\<^sup>+ x. f (T x) \<partial>M"
integration_by_parts_interior: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'b"   and y :: "'c" assumes "bounded_bilinear prod"   and "a \<le> b"   and "continuous_on {a..b} f"   and "continuous_on {a..b} g"   and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"   and "((\<lambda>x. prod (f x) (g' x)) has_integral prod (f b) (g b) - prod (f a) (g a) - y) {a..b}" shows "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
bind_gpv_assoc: fixes f :: "('h, 'e, 'f) gpv"   and g :: "'h \<Rightarrow> ('g, 'e, 'f) gpv"   and h :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "f \<bind> g \<bind> h = f \<bind> (\<lambda>x. g x \<bind> h)"
search_nonempty_eq0_1: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and i :: "nat"   and y :: "nat" assumes "total_cons s"   and "s b \<down>= i"   and "\<phi> i y \<down>"   and "e_length b \<le> y" shows "eval (total_cons.r_search s) [b] \<down>"
respects_unit: fixes a :: "'a" assumes "C.obj a" shows "(\<tau>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0 a] = \<tau>\<^sub>1 a \<cdot>\<^sub>D (G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a)"
prems: shows "is_tangle_diagram (z * zs) \<and> codomain_wall (z * zs) = domain_wall y"
inres_simps_3: fixes x :: "'c" shows "inres (RETURN x) = (=) x"
select_convs_2: fixes pverts :: "'a set"   and parcs :: "('a \<times> 'a) set"   and more :: "'b" shows "parcs \<lparr>pverts = pverts, parcs = parcs, \<dots> = more\<rparr> = parcs"
rep_eq: fixes x :: "(nat, 'd) slist" shows "map_of_slist x = map_of (Rep_slist x)"
floor_nat_add_id: fixes r :: "real"   and n :: "nat" assumes "0 \<le> r"   and "r < 1" shows "\<lfloor>real n + r\<rfloor> = int n"
ap_fv_subset: fixes S :: "stack_elem list" shows "ap S \<subseteq> fv S"
B_def: shows "B \<equiv> \<lambda>m. disjointed (A m)"
C1: fixes Q :: "('a, 'b, 'c) psi" shows "(\<Psi>, Q \<parallel> \<zero>, Q) \<in> Rel"
integrable_sum: fixes I :: "'d set"   and M :: "'e measure"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
dgrad_p_set_subset: fixes m :: "nat"   and n :: "nat"   and d :: "'a \<Rightarrow> nat" assumes "m \<le> n" shows "punit.dgrad_p_set d m \<subseteq> punit.dgrad_p_set d n"
hyps: shows "supportE (\<lambda>l. preTz C (e l) n) \<subseteq> supportE e"
map_try_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf"   and q :: "'e spmf" shows "map_spmf f (TRY p ELSE q) = TRY map_spmf f p ELSE map_spmf f q"
memory_write_low_equal2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and addr :: "word32"   and bm :: "word4"   and data :: "word32" assumes "low_equal s1 s2 \<and> Some t1 = memory_write 10 addr bm data s1" shows "\<exists>t2. Some t2 = memory_write 10 addr bm data s2"
insert_compr: fixes a :: "'a"   and B :: "'a set" shows "a \<triangleright> B = {x. x = a \<or> x \<in> B}"
complex_surj: fixes z :: "complex" shows "Complex (Re z) (Im z) = z"
right_inverse_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(divide a b = one) = (a = b)"
enn2real_less_iff: fixes x :: "ennreal"   and c :: "real" assumes "x < top" shows "(enn2real x < c) = (x < ennreal c)"
mk_mat_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" shows "HMA_Connect.mk_mat f = (\<chi>i j. f i j)"
quot_exhaust: fixes A :: "'d quot" shows "\<exists>a. A = \<lfloor>a\<rfloor>"
mset_le_incr_right2: fixes a :: "'f multiset"   and b :: "'f multiset"   and c :: "'f multiset" assumes "a \<subseteq># b" shows "a \<subseteq># c + b"
wlconf_def: fixes G :: "prog"   and s :: "st"   and vs :: "'a \<Rightarrow> val option"   and Ts :: "'a \<Rightarrow> ty option" shows "G,s\<turnstile>vs[\<sim>\<Colon>\<preceq>]Ts = (\<forall>n. ! T:Ts n: ! v:vs n: G,s\<turnstile>v\<Colon>\<preceq>T)"
ord_of_nat_mono_iff: fixes i :: "nat"   and j :: "nat" shows "(ord_of_nat i \<le> ord_of_nat j) = (i \<le> j)"
compl: shows "\<forall>x\<in>carrier. \<forall>y\<in>carrier. x \<succeq> y \<or> y \<succeq> x"
assms_2: shows "P \<Longrightarrow>\<^sub>\<tau> P'"
assms_1: shows "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
emeasure_eq_ennreal_measure: fixes M :: "'b measure"   and A :: "'b set" assumes "emeasure M A \<noteq> top" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
observable_transition_stepI: fixes P :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" assumes "P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle>" shows "P \<Rightarrow>{\<alpha>}\<^sub>S P'"
step: fixes x :: "hf" assumes "Ord x"   and "\<And>y. y \<^bold>\<in> x \<Longrightarrow> P y" shows "P x"
abs_diff_less_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and a :: "'c"   and r :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (abs (minus x a)) r = (less (minus a r) x \<and> less x (plus a r))"
exhaust: fixes y :: "'a node_action" obtains x11 :: "nat set" and x12 :: "'a" where "y = x11:*cast(x12)"   | x21 :: "nat" and x22 :: "nat" where "y = x21:deliver(x22)"   | x31 :: "nat set" and x32 :: "nat set" and x33 :: "'a" where "y = x31\<not>x32:arrive(x33)"   | x41 :: "nat" and x42 :: "nat" where "y = connect(x41, x42)"   | x51 :: "nat" and x52 :: "nat" where "y = disconnect(x51, x52)"   | x61 :: "nat" and x62 :: "nat" and x63 :: "nat" where "y = x61:newpkt(x62, x63)"   | "y = \<tau>\<^sub>n"
map_of_Cons_code_2: fixes l :: "'m"   and v :: "'l"   and ps :: "('m \<times> 'l) list"   and k :: "'m" shows "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
init_fin_\<tau>move_simps_3: fixes x :: "'x2"   and m :: "'m2"   and ta :: "('l, 't, status \<times> 'x2, 'm2, 'w, 'o action) thread_action"   and s :: "status"   and x' :: "'x2"   and m' :: "'m2" shows "r2.init_fin_\<tau>move ((Running, x), m) ta ((s, x'), m') = (\<exists>ta'. ta = convert_TA_initial (convert_obs_initial ta') \<and> s = Running \<and> \<tau>move2 (x, m) ta' (x', m'))"
pr0_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "pr0 a b = (if ide a \<and> ide b then mkArr (local.set (local.prod a b)) (local.set b) (\<lambda>x. UP (hsnd (DOWN x))) else null)"
f_shrink_last_Cons: fixes k :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "0 < k"   and "length xs = k" shows "(xs @ ys) \<div>\<^sub>l k = last xs # ys \<div>\<^sub>l k"
generalizes_lit_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and L :: "'b literal"   and M :: "'b literal" shows "substitution_ops.generalizes_lit subst_atm L M = (\<exists>\<sigma>. substitution_ops.subst_lit subst_atm L \<sigma> = M)"
irreducible_dvd_prod: fixes p :: "'c poly"   and f :: "'d \<Rightarrow> 'c poly"   and as :: "'d set" assumes "irreducible p"   and "p dvd prod f as" shows "\<exists>a\<in>as. p dvd f a"
zero_adds_pp: fixes v :: "'t" shows "(0::'a) adds\<^sub>p v"
trm_subst_ident': fixes v :: "'a"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<notin> subst_domain \<theta>" shows "Var v \<cdot> \<theta> = Var v"
toplevel_inters_in_lang: fixes w :: "'a list"   and n :: "nat"   and s :: "'b rexp" shows "(w \<in> lang n s) = (\<forall>r\<in>toplevel_inters s. w \<in> lang n r)"
bv_sadd_type1: fixes w1 :: "bit list"   and w2 :: "bit list" shows "bv_sadd (norm_signed w1) w2 = bv_sadd w1 w2"
inducts: fixes g :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'a \<times> 'c)"   and x :: "'c"   and Pa :: "'c \<Rightarrow> bool" assumes "local.filter_has_next g x"   and "\<And>s. \<lbrakk>fst g s; P (fst (snd g s))\<rbrakk> \<Longrightarrow> Pa s"   and "\<And>s. \<lbrakk>fst g s; \<not> P (fst (snd g s)); local.filter_has_next g (snd (snd g s)); Pa (snd (snd g s))\<rbrakk> \<Longrightarrow> Pa s" shows "Pa x"
dim_eq_span: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "dim T \<le> dim S" shows "span S = span T"
div_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (div_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
"2_1": fixes x :: "'a"   and xa :: "'a list" assumes "x = fold min_for zs z"   and "xa = remove1 x (z # zs)"   and "set xa \<subseteq> S" shows "sortedP (selsort xa)"