hcju/isabelleps huggingface dataset:
Pair_def: fixes a :: "'a"   and b :: "'b" shows "(a, b) = Abs_prod (Pair_Rep a b)"
h1_2: shows "r1 = h1 \<otimes> g"
rel_compp: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_rexp (R OO S) = rel_rexp R OO rel_rexp S"
that: fixes f :: "'a \<Rightarrow> 'b" assumes "Y = f ` X"   and "\<forall>x\<in>X. R x (f x)" shows "thesis"
n_kat_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "t x \<cdot> y \<cdot> t z = y \<cdot> t z" shows "!x \<cdot> y \<cdot> !z = !x \<cdot> y"
Abs_bd_type_F_inject: fixes x :: "bd_type_F1 + bd_type_F2"   and y :: "bd_type_F1 + bd_type_F2" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Abs_bd_type_F x = Abs_bd_type_F y) = (x = y)"
chart_eucl_simps_1: shows "domain chart_eucl = UNIV"
rep_eq: fixes x :: "'d \<Longrightarrow> 'c" shows "\<lbrakk>x\<rbrakk>\<^sub>e = get\<^bsub>x\<^esub>"
content_triangle: fixes A :: "(real, 2) vec"   and B :: "(real, 2) vec"   and C :: "(real, 2) vec" shows "content (convex hull {A, B, C}) = \<bar>(C $ 1 - A $ 1) * (B $ 2 - A $ 2) - (B $ 1 - A $ 1) * (C $ 2 - A $ 2)\<bar> / 2"
nprv_clear4_2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>3 :: "'c"   and \<phi>4 :: "'c"   and \<psi> :: "'c"   and \<phi>2 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>3, \<phi>4} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3, \<phi>4} \<psi>"
simple_function_cong_algebra: fixes N :: "'b measure"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "sets N = sets M"   and "space N = space M" shows "simple_function M f = simple_function N f"
Legendre_3_M: shows "Legendre 3 (int M) = - 1"
assms: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'a" shows "f \<cdot> (g \<cdot> h) = f \<cdot> g \<cdot> h"
outputPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "(P, Q) \<in> Rel" shows "a{b}.P \<leadsto>\<^sup>^<Rel> a{b}.Q"
simps_2: fixes uu :: "'a" shows "- ext0.Val uu = ext0.Bot"
functor_axioms: shows "functor F.C.CCC.comp B F.C.T.ToTC"
distproj_in_proj_set_1: fixes x :: "'b"   and A :: "'b set" assumes "proj_set x A \<noteq> {}" shows "distproj x A \<in> proj_set x A"
lens_equiv_sym: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "X \<approx>\<^sub>L Y" shows "Y \<approx>\<^sub>L X"
c: fixes P :: "'a \<Rightarrow> ('b \<times> 'b) set" assumes "Pcond A Is P" shows "complete A (scf P)"
is_ascii_fresh_string: fixes Y :: "char list"   and Xs :: "char list set" assumes "is_ascii Y" shows "is_ascii (fresh_string Xs Y)"
kstar_unfoldr_eq: fixes f :: "'a \<Rightarrow> 'a set" shows "\<eta> \<squnion> kstar f \<circ>\<^sub>K f = kstar f"
mem_simps_8: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
B_def: shows "B \<equiv> {b \<in> A. \<forall>a\<in>X. a \<sqsubseteq> b}"
subterms_Unop: shows "subterms Unop = {Unop}"
is_root_imp_pdivides: fixes p :: "'a list"   and x :: "'a" assumes "p \<in> carrier (poly_ring R)"   and "is_root p x" shows "[\<one>, \<ominus> x] pdivides p"
id_o: fixes g :: "'a \<Rightarrow> 'b" shows "id \<circ> g = g"
std_dual_def: fixes a :: "'a" shows "local.std_dual a = restrict0 S (restrict0 S (local.inner_Basis a))"
symmetric_Q: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q I a b" shows "Q I b a"
im_diff: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and g :: "'b"   and g' :: "'b" assumes "RModuleHom R smult M smult' T"   and "g \<in> M"   and "g' \<in> M" shows "T (g - g') = T g - T g'"
lookup_env_restr_eq: fixes S :: "'e set"   and m :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(m f|` S) x = (if x \<in> S then m x else \<bottom>)"
ltl_to_rabin_correct_exposed: fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "ltl_FG_to_rabin \<Sigma> \<G> w" shows "\<PP>\<^sub>\<infinity> \<phi> \<G> w = accept\<^sub>R (semi_mojmir_def.step \<Sigma> \<up>af\<^sub>G (Abs \<phi>), semi_mojmir_def.initial (Abs \<phi>), {(mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<up>af\<^sub>G (Abs \<phi>) {q. \<G> \<Turnstile>\<^sub>P Rep q} \<union> mojmir_to_rabin_def.merge\<^sub>R \<up>af\<^sub>G (Abs \<phi>) {q. \<G> \<Turnstile>\<^sub>P Rep q} i, mojmir_to_rabin_def.succeed\<^sub>R \<up>af\<^sub>G (Abs \<phi>) {q. \<G> \<Turnstile>\<^sub>P Rep q} i) |i. i < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs \<phi>)}) w"
bmf: fixes k :: "nat" shows "(\<lambda>x. (if x \<le> a + 1 / real (Suc k) then real (Suc k) * d * (a + 1 / real (Suc k) - x) else 0)\<^sup>2) \<in> borel_measurable (lebesgue_on {a..b})"
iMODb_imirror_bounds: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and l :: "nat"   and r :: "nat" assumes "r' + m' * c' \<le> l + r" shows "imirror_bounds [ r', mod m', c' ] l r = [ l + r - r' - m' * c', mod m', c' ]"
nilRes: fixes x :: "name" shows "<\<nu>x>\<zero> \<sim> \<zero>"
A: shows "\<guillemotleft>\<l>[f] : trg f \<star> f \<Rightarrow> f\<guillemotright>"
closure_of_subtopology_mono: fixes T :: "'c set"   and U :: "'c set"   and X :: "'c topology"   and S :: "'c set" assumes "T \<subseteq> U" shows "subtopology X T closure_of S \<subseteq> subtopology X U closure_of S"
IH: fixes m\<^sub>1 :: "('a, 'b) fmap" assumes "fmdom' m\<^sub>1 \<inter> fmdom' m\<^sub>2 = {}" shows "m\<^sub>1 ++\<^sub>f m\<^sub>2 = m\<^sub>2 ++\<^sub>f m\<^sub>1"
d\<^sub>\<kappa>_def: shows "d\<^sub>\<kappa> \<equiv> map_fun eval\<kappa> id id"
tv_id_1: fixes A :: "ty" shows "T (V A) \<approx> A"
eval_wfp: fixes p :: "prog"   and \<E> :: "char list \<Rightarrow> nat option" assumes "wfp p"   and "\<E> \<in> envs p" shows "eval \<E> (body p) \<noteq> None"
Jacobi_eq_0_iff_not_coprime: fixes p :: "int"   and n :: "int" assumes "p \<noteq> 0"   and "p \<noteq> 1" shows "(Jacobi n p = 0) = (\<not> coprime n p)"
2: fixes m :: "nat"   and n :: "nat" assumes "x [^] m = x [^] n"   and "\<forall>i j. x [^] i = x [^] j \<longrightarrow> i = j" shows "m = n"
ln_add_one_self_less_self: fixes x :: "real" assumes "0 < x" shows "ln (1 + x) < x"
dividesE: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "p1 divides p2" obtains q :: "'a list" where "poly p2 = poly (p1 *** q)"
ereal_SUP: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "\<bar>SUP a\<in>A. ereal (f a)\<bar> \<noteq> \<infinity>" shows "ereal (Sup (f ` A)) = (SUP a\<in>A. ereal (f a))"
LeftDerivationLadder_cut_appendix_helper: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder (\<alpha> @ \<delta>) D L \<gamma>"   and "ladder_i L 0 < length \<alpha>" shows "\<exists>E F \<gamma>1 \<gamma>2 L'. D = E @ F \<and> \<gamma> = \<gamma>1 @ \<gamma>2 \<and> LeftDerivationLadder \<alpha> E L' \<gamma>1 \<and> derivation_ge F (length \<gamma>1) \<and> LeftDerivation \<delta> (derivation_shift F (length \<gamma>1) 0) \<gamma>2 \<and> L' = ladder_cut L (length E)"
under_trans: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "trans r"   and "a \<in> under r b"   and "b \<in> under r c" shows "a \<in> under r c"
sum_list'_def: fixes p :: "nat"   and xs :: "'b list" shows "sum_list' p xs = fold (\<lambda>a b. eucl_truncate_up p (a + b)) xs (0::'b)"
aff_dim_UNIV: shows "aff_dim UNIV = int DIM('b)"
eq: shows "\<integral>\<^sup>+x\<in>A. ennreal (f x)\<partial>lborel = \<integral>\<^sup>+ x. ennreal (indicat_real A x * f x) \<partial>lborel"
ord_dual: fixes dual :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ord_with_dual dual less_eq"   and "less_eq x y" shows "less_eq (dual y) (dual x)"
left_closed: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E" shows "a \<in> S"
simp_snd_lmap: shows "lhd (lmap snd Ns) = snd (lhd Ns)"
induced_graph_imp_symmetric: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "symmetric G"   and "induced_subgraph H G" shows "symmetric H"
isRecvMarker_def: fixes event :: "('d, 'e, 'f) event" shows "isRecvMarker event = (\<exists>x51 x52 x53. event = RecvMarker x51 x52 x53)"
isomorphic_implies_same_tab_1: fixes r' :: "'a" assumes "r \<cong> r'" shows "tab\<^sub>0 r = tab\<^sub>0 r'"
simps_3: fixes va :: "'b \<times> nat"   and vb :: "('b \<times> nat) tree" shows "skew \<langle>\<langle>\<rangle>, va, vb\<rangle> = \<langle>\<langle>\<rangle>, va, vb\<rangle>"
polynomial_f_def: fixes rs :: "nat list"   and y :: "nat"   and shared_weights :: "bool"   and w :: "nat \<Rightarrow> real" assumes "deep_model_correct_params_y rs y" shows "deep_model_correct_params_y.polynomial_f shared_weights rs y w = det (submatrix (ten2mat (deep_model_correct_params_y.A shared_weights rs y w)) (deep_model_correct_params_y.rows_with_1 rs y) (deep_model_correct_params_y.rows_with_1 rs y))"
prod_empt_def: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list" shows "prod_empt A B = (replicate (length (fst A)) (replicate (length (fst B)) None), [])"
lfinite_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lfinite (lappend xs ys) = (lfinite xs \<and> lfinite ys)"
butlast_zip_cons: fixes ws :: "'d list"   and r1s :: "'e list"   and w :: "'d"   and r1 :: "'e" assumes "length ws = length r1s"   and "ws \<noteq> []" shows "butlast (w # ws || r1 # r1s) = (w, r1) # butlast (ws || r1s)"
dvd_period: fixes a :: "int"   and d :: "int"   and x :: "int"   and t :: "int"   and c :: "int" assumes "a dvd d" shows "(a dvd x + t) = (a dvd x + c * d + t)"
minus_interval_def: fixes a :: "'b interval"   and b :: "'b interval" shows "a - b = a + - b"
error_def: fixes x :: "'g" shows "error x = sum_bot.Left x"
pred_envir_conv: fixes p :: "'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e" shows "pred_envir p f = (\<forall>x. p (f x))"
degree_pderiv_le: fixes p :: "'b poly" shows "degree (pderiv p) \<le> degree p - 1"
arr_char: fixes f :: "'a" shows "S.arr f = right a f"
eval_fm_Special_axP: fixes e :: "(name, hf) finfun"   and p :: "tm" shows "eval_fm e (Special_axP p) = (\<lbrakk>p\<rbrakk>e \<in> Special_ax)"
higher_deriv_mult: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and z :: "complex"   and n :: "nat" assumes "f holomorphic_on S"   and "g holomorphic_on S"   and "open S"   and "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f w * g w) z = (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
LIMSEQ_SUP: fixes X :: "nat \<Rightarrow> 'a" assumes "incseq X" shows "X \<longlonglongrightarrow> Sup (range X)"
"1_1": shows "\<forall>ys. length ys < length xs \<longrightarrow> ys \<noteq> [] \<longrightarrow> toplevel_summands (derivs ys (Star r)) \<subseteq> {Times (derivs ysa r) (Star r) |ysa. \<exists>zs. ysa \<noteq> [] \<and> zs @ ysa = ys}"
intros_12: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec2"   and "g \<in> PrimRec1"   and "h \<in> PrimRec1" shows "(\<lambda>x. f (g x) (h x)) \<in> PrimRec1"
le_less: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(x \<le>s y) = (x <s y \<or> x = y)"
ideal_def: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" shows "ideal I R \<equiv> additive_subgroup I R \<and> ring R \<and> ideal_axioms I R"
rreq_rrep_fresh_1: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool" shows "rreq_rrep_fresh crt (Rreq hops rreqid dip dsn dsk oip osn sip handled) = (sip \<noteq> oip \<longrightarrow> oip \<in> kD crt \<and> (osn < sqn crt oip \<or> sqn crt oip = osn \<and> the (dhops crt oip) \<le> hops \<and> the (flag crt oip) = val))"
Y: fixes x :: "'a"   and y :: "'a" assumes "perm_restrict f (S - A) x = perm_restrict f (S - A) y" shows "x = y"
B: shows "prv (imp (lcnj \<phi>1s) (lcnj \<phi>2s'))"
5: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
upper_split_interval1: fixes X :: "'b interval"   and m :: "'b" shows "upper (fst (split_interval X m)) = max (lower X) m"
less_asym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y"   and "\<not> P \<Longrightarrow> less y x" shows "P"
mem_hmsetmset_imp_less: fixes M :: "hmultiset"   and N :: "hmultiset" assumes "M \<in># hmsetmset N" shows "M < N"
r_enumdom_recfn: shows "recfn 2 r_enumdom"
funcset_mem: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'g set"   and x :: "'f" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
shrK_in_knows: fixes A :: "agent"   and evs :: "event list" shows "Key (shrK A) \<in> knows A evs"
fmfilter_alt_defs_4: fixes A :: "'c set" shows "fmrestrict_set A = fmfilter (\<lambda>a. a \<in> A)"
landau_function_family_axioms: shows "landau_function_family F G"
ipresIGSwapAbs_def: fixes hA :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'k, 'c, 'l) model_scheme"   and MOD' :: "('m, 'n, 'g, 'o, 'p, 'j, 'q, 'd, 'r) model_scheme" shows "ipresIGSwapAbs hA MOD MOD' \<equiv> \<forall>zs z1 z2 us s A. igWlsAbs MOD (us, s) A \<longrightarrow> hA (igSwapAbs MOD zs z1 z2 A) = igSwapAbs MOD' zs z1 z2 (hA A)"
red_plus_cs: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" shows "relation.cs (red F) (p + r) (q + r)"
NF_not_suc: fixes x :: "'d"   and y :: "'d"   and r :: "('d \<times> 'd) set" assumes "(x, y) \<in> r\<^sup>*"   and "x \<in> NF r" shows "x = y"
no_outgoing_transitions: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" assumes "ffilter (\<lambda>((s', uu_), uu_). s = s') e = {||}" shows "possible_steps e s r l i = {||}"
permut_aux_sound: fixes i :: "nat"   and as :: "'b list"   and f :: "nat \<Rightarrow> nat"   and bs :: "'b list" assumes "i < length as" shows "permut_aux as f bs ! i = bs ! f i"
measurable_sets_borel: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'b measure"   and A :: "'b set" assumes "f \<in> borel \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "f -` A \<in> sets borel"
seq_conv_if: fixes x :: "'s" shows "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)"
rt: fixes x :: "'a" shows "poly p x \<noteq> (0::'a)"
cidr_split_prefix: fixes r :: "'a wordinterval" shows "\<Union> (uncurry ipset_from_cidr ` set (cidr_split r)) = wordinterval_to_set r"
filter_paths_conv_free_restr2: fixes S :: "'a set"   and t :: "'a ttree" shows "filter (\<lambda>x'. x' \<notin> S) ` paths t = paths (ttree_restr (- S) t)"
filter_id_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
assms_2: shows "cosh (f (Lim F (\<lambda>x. x))) \<noteq> (0::'a)"
out: fixes x :: "'v vertex" assumes "x \<in> \<^bold>V\<^bsub>\<Delta>''\<^esub>"   and "x \<noteq> source \<Delta>''" shows "d_OUT (capacity \<Delta>'') x < \<top>"
chain_ord_spmf_eqD: shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>p x. ennreal (spmf p x)) ` Y)"
Chain_Torder: fixes X :: "'a set" assumes "Chain D X" shows "Torder (Iod D X)"
eventually_prod_filter: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and F :: "'b filter"   and G :: "'c filter" shows "eventually P (F \<times>\<^sub>F G) = (\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))"
bit_b''2: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' BIT s qs = 75 / 10\<^sup>2 * real (length qs - 1) - 5 / 10"
C: shows "P''' \<Longrightarrow>\<^sub>\<tau> P'"
elims: fixes x :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_uminus x = y" obtains a :: "complex" and b :: "complex" and c :: "complex" and d :: "complex" where "x = (a, b, c, d)" and "y = (- a, - b, - c, - d)"
HPhase1or2ReadThen_HInv4a_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4a s p"   and "Inv2b s" shows "HInv4a s' p"
zeroA_def: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and z1 :: "'e" shows "zeroA z i f A z1 = (\<forall>x\<in>addition_set f (aug_pm_set z i A). z1 \<^sub>f+ x = x)"
inext_neq_iMin_infin: fixes I :: "nat set"   and n :: "nat" assumes "infinite I" shows "inext n I \<noteq> iMin I"
simps_2: fixes S :: "'a set"   and n :: "nat" shows "f S (Suc n) = least S # f (S - {least S}) n"
simps_8: fixes f :: "'e \<Rightarrow> 'd" shows "map_option f \<bottom> = \<bottom>"
degree0_coeffs: fixes p :: "'d poly" assumes "degree p = 0" shows "\<exists>a. p = [:a:]"
more_temp_simps1_3: fixes F :: "'a \<Rightarrow> bool" shows "\<not> \<box>F \<equiv> \<diamond>\<not> F"
intro: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" assumes "ltl_equivalence eq"   and "af_congruent_axioms eq" shows "af_congruent eq"
Fun_ide: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "Fun a = (\<lambda>x\<^bold>\<in>local.set a. x)"
splitAt_take: fixes ls :: "'c list"   and i :: "nat" assumes "distinct ls"   and "i < |ls|" shows "fst (splitAt (ls ! i) ls) = take i ls"
concat_chop: fixes n :: "nat"   and xs :: "'b list" assumes "0 < n" shows "concat (chop n xs) = xs"
trans_closure: fixes M1 :: "lam"   and M2 :: "lam" shows " M1 \<longrightarrow>\<^sub>1\<^sup>* M2 =  M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2"
map_of_append: fixes xs :: "('d \<times> 'e) list"   and ys :: "('d \<times> 'e) list" shows "map_of (xs @ ys) = map_of ys ++ map_of xs"
intros_9: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> \<sigma>" shows "Sequence \<tau> \<sqsubset> Sequence \<sigma>"
cSup_le_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set"   and a :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "S \<noteq> {}"   and "preorder.bdd_above less_eq S" shows "less_eq (Sup S) a = (\<forall>x\<in>S. less_eq x a)"
good_qAbs_pick_X: shows "qGoodAbs (qAbs xs x (pick X))"
succ_chain_bounds_id: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "\<B>\<^sub>i (l ! i) = \<B>\<^sub>i (l ! j)"
Collect_conv_if: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "{x. x = a \<and> P x} = (if P a then {a} else {})"
step_reachable: fixes x :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and y :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" assumes "step x = y" shows "reachable x y"
zassenhaus_reconstruction_def: fixes vs :: "int poly list"   and p :: "int"   and n :: "nat"   and f :: "int poly" shows "zassenhaus_reconstruction vs p n f = (let mul = poly_mod.mul_const (p ^ n); sl_impl = my_subseqs.impl (\<lambda>x. map_prod (mul x) ((#) x)) in zassenhaus_reconstruction_generic sl_impl vs p n f)"
norm: shows "\<forall>a\<in>atoms \<phi>. divisor a \<noteq> 0"
R_Qi_Aj: fixes A :: "((real, 'a) vec, 'b) vec"   and i :: "'a"   and j :: "'a" shows "snd (QR_decomposition A) $ i $ j = column i (fst (QR_decomposition A)) \<bullet> column j A"
characteriseAx: fixes r :: "'a list \<times> 'b sequent" assumes "r \<in> Ax" shows "r = ([],  \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) \<or> (\<exists>i. r = ([],  \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>))"
strict_coboundedI2: fixes c :: "'b"   and b :: "'b"   and a :: "'b" assumes "c < b" shows "c < max a b"
remdups_eq_map_eq: fixes xa :: "'b list"   and ya :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "remdups xa = remdups ya" shows "remdups (list.map f xa) = remdups (list.map f ya)"
OclIncluding_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Sequence('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "x->including\<^sub>S\<^sub>e\<^sub>q(y) = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>\<lceil>\<lceil>Rep_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> @ [y \<tau>]\<rfloor>\<rfloor> else invalid \<tau>)"
ih_25: shows "InvariantVarsF (getF state') F0 Vbl"
T_slow_median_def: fixes xs :: "'b list" shows "T_slow_median xs = T_slow_select ((length xs - 1) div 2) xs + 1"
map_comp0: fixes g1 :: "'s \<Rightarrow> 'q"   and f1 :: "'o \<Rightarrow> 's"   and g2 :: "'t \<Rightarrow> 'r"   and f2 :: "'p \<Rightarrow> 't" shows "Fmap (g1 \<circ> f1) (g2 \<circ> f2) = Fmap g1 g2 \<circ> Fmap f1 f2"
loop_iterate_axiom_def: shows "loop_iterate_axiom \<equiv> [[$\<alpha> vid1**]]Pc pid1 \<leftrightarrow> (Pc pid1 && [[$\<alpha> vid1]][[$\<alpha> vid1**]]Pc pid1)"
closed_segment_iv_subset_domain: fixes t :: "real" assumes "t \<in> cball t__ et" shows "{t__--t} \<subseteq> cball t__ et"
b_a_div_s: shows "(b - a) div m = ((b div m - a div m) * m + b mod m - a mod m) div m"
not_irreducible_zero: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.irreducible times one zero zero"
stat_ref3: shows "statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"