hcju/isabelleps huggingface dataset:
less_extended_def: shows "(<) \<equiv> rec_extended (case_extended False (\<lambda>y_0. True) True) (\<lambda>x_0. case_extended False ((<) x_0) True) (case_extended False (\<lambda>y_0. False) False)"
integer_hull_mono: fixes P :: "'a vec set"   and Q :: "'a vec set" assumes "P \<subseteq> Q" shows "integer_hull P \<subseteq> integer_hull Q"
rpf_g: fixes n :: "nat" shows "real_polynomial_function (g n)"
connected: shows "\<forall>v\<in>V. \<forall>v'\<in>V. nodes_connected G v v'"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C E.FG.map E.FG.cmp"
Int_absorb2: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A" shows "A \<inter> B = A"
\<Gamma>_Var_fst': fixes \<tau> :: "('fun, 'atom) Term.term"   and n :: "nat"   and m :: "nat" shows "\<Gamma> (Var (\<tau>, n)) = \<Gamma> (Var (\<tau>, m))"
rreq_rrep_fresh_def: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and m :: "msg" shows "rreq_rrep_fresh crt m \<equiv> case m of Rreq hopsc x xa xb xc oipc osnc ipcc \<Rightarrow> ipcc \<noteq> oipc \<longrightarrow> oipc \<in> kD crt \<and> (osnc < sqn crt oipc \<or> sqn crt oipc = osnc \<and> the (dhops crt oipc) \<le> hopsc \<and> the (flag crt oipc) = val) | Rrep hopsc dipc dsnc x ipcc \<Rightarrow> ipcc \<noteq> dipc \<longrightarrow> dipc \<in> kD crt \<and> sqn crt dipc = dsnc \<and> the (dhops crt dipc) = hopsc \<and> the (flag crt dipc) = val | _ \<Rightarrow> True"
da_shunt1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> d y + z" shows "x \<cdot> ad y \<le> z"
assms: shows "\<forall>i dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> rt (\<sigma> i) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)"
impl_Impl: fixes x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg" shows "implem (Chan x A B M) \<notin> payload"
tSplice_spr_jview_a: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "tObsC t = tObsC t'" shows "spr_jview a (t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t') = spr_jview a t'"
smult_assoc2: fixes a :: "'a"   and x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "x \<in> carrier P"   and "y \<in> carrier P" shows "a \<odot>\<^bsub>P\<^esub> x \<otimes>\<^bsub>P\<^esub> y = a \<odot>\<^bsub>P\<^esub> (x \<otimes>\<^bsub>P\<^esub> y)"
cinner_diff_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<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 dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (minus x y) z = cinner x z - cinner y z"
proj_list_insert_1: fixes i :: "'a"   and d :: "'b"   and D :: "('a strand_label \<times> 'b) list" shows "proj i (List.insert (ln i, d) D) = List.insert (ln i, d) (proj i D)"
pcoef_eq: shows "\<forall>x\<in>vars (apps (Hd \<zeta>) ss). \<forall>i. A (PCoef x i) = B (PCoef x i)"
monomial_is_monomial: fixes c :: "'d"   and t :: "'e" assumes "c \<noteq> (0::'d)" shows "is_monomial (monomial c t)"
ptensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "b.ptensor_mat m1 m2 = tensor_mat b.dims0 b.vars1' m1 m2"
assms_6: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq A B"
subsumes_modulo_def: fixes C :: "('b, 'c) Term.term literal multiset"   and D :: "('b, 'c) Term.term literal multiset"   and \<sigma> :: "'c \<Rightarrow> ('b, 'c) Term.term option" shows "subsumes_modulo C D \<sigma> = (\<exists>\<tau>. dom \<tau> = vars_clause C \<union> dom \<sigma> \<and> extends_subst \<sigma> \<tau> \<and> subst_cls C (subst_of_map Var \<tau>) \<subseteq># D)"
signs: fixes i :: "nat"   and x :: "real" assumes "i < length (p # ps) - 2"   and "poly ((p # ps) ! (i + 1)) x = 0" shows "poly ((p # ps) ! (i + 2)) x * poly ((p # ps) ! i) x < 0"
subspace_UNIV: fixes p :: "'c" shows "subspace (manifold_eucl.dest.tangent_space p)"
sep_conj_false_right: fixes P :: "'a \<Rightarrow> bool" shows "(P \<and>* (\<lambda>s. False)) = (\<lambda>s. False)"
pwd_spy_id_pubk_def: fixes n :: "nat" shows "pwd_spy_id_pubk n \<equiv> case pwd_spy_id_prik n of (s, S, A, B, C, D) \<Rightarrow> (insert (Spy, \<langle>n, PubKey S\<rangle>) s, S, A, B, C, D)"
dbm_entry_val_mono_1: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and c' :: "'c"   and b :: "'d DBMEntry"   and b' :: "'d DBMEntry" assumes "dbm_entry_val u (Some c) (Some c') b"   and "b \<preceq> b'" shows "dbm_entry_val u (Some c) (Some c') b'"
distrib: fixes g :: "'c \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. g x + h x) = sum g A + sum h A"
document_ptr_kinds_eq_h: shows "document_ptr_kinds h2 = document_ptr_kinds h"
cases_scheme: fixes r :: "'b control_state_scheme" obtains break_status :: "bool" and return_status :: "bool" and more :: "'b" where "r = \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr>"
g_add_dj_def: fixes ops :: "('b, 'c, 'a, 'd) map_basic_ops_scheme"   and m1 :: "'a"   and m2 :: "'a" shows "StdBasicMapDefs.g_add_dj ops m1 m2 \<equiv> poly_map_iteratei_defs.iteratei (bmap_op_list_it ops) m2 (\<lambda>_. True) (\<lambda>(k, v). bmap_op_update_dj ops k v) m1"
UNC_def: fixes A :: "('b \<times> 'b) set" shows "UNC A = (\<forall>a b. a \<in> NF A \<and> b \<in> NF A \<and> (a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>* \<longrightarrow> a = b)"
domain_atlas_subset_carrier: fixes c :: "('a, 'b) chart" assumes "c \<in> atlas" shows "domain c \<subseteq> carrier"
elims_3: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('i, 'f, 'j, 'k) SLin_state_scheme" assumes "\<not> P6 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('i, 'f, 'j, 'k) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (\<forall>p. (status s1 p \<noteq> Aborted) = (status s2 p = Sleep))"
fin_digraph_axioms_def: fixes G :: "('c, 'd) pre_digraph" shows "fin_digraph_axioms G \<equiv> finite (verts G) \<and> finite (arcs G)"
BAD_iff: fixes f :: "nat \<Rightarrow> 'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(f \<in> mbs.BAD A P) = ((\<forall>i. f i \<in> A) \<and> bad P f)"
call_of_return_node_def: fixes n :: "'node"   and n' :: "'node" shows "call_of_return_node n n' \<equiv> \<exists>a a'. return_node n \<and> valid_edge a \<and> n' = sourcenode a \<and> valid_edge a' \<and> a' \<in> get_return_edges a \<and> n = targetnode a'"
partition_on_Diff: fixes A :: "'b set"   and P :: "'b set set"   and Q :: "'b set set" assumes "partition_on A P"   and "Q \<subseteq> P" shows "partition_on (A - \<Union> Q) (P - Q)"
divmod_digit_0_2: 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 division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and b :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step"   and "less zero b"   and "less (modulo a (times (numeral.numeral one plus (num.Bit0 num.One)) b)) b" shows "modulo a (times (numeral.numeral one plus (num.Bit0 num.One)) b) = modulo a b"
headnz: fixes p :: "poly"   and n :: "nat"   and m :: "nat" assumes "isnpolyh p n"   and "p \<noteq> 0\<^sub>p" shows "headn p m \<noteq> 0\<^sub>p"
inject: fixes x31 :: "ty_Rep"   and x32 :: "ty_Rep"   and y31 :: "ty_Rep"   and y32 :: "ty_Rep" shows "(Arrow_Rep x31 x32 = Arrow_Rep y31 y32) = (x31 = y31 \<and> x32 = y32)"
prefix_Nil: fixes xs :: "'e list" shows "prefix xs [] = (xs = [])"
assms_3: fixes x :: "'a" assumes "x \<in> space M" shows "norm (f x) \<le> g x"
col_cong2_bet1: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Col A B D"   and "Bet A C B"   and "Cong A B C D"   and "Cong A C B D" shows "Bet C B D"
tendsto_iff_Liminf_eq_Limsup: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd"   and f0 :: "'d" assumes "F \<noteq> bot" shows "(f \<longlongrightarrow> f0) F = (Liminf F f = f0 \<and> Limsup F f = f0)"
rep_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "'d \<Rightarrow>\<^sub>0 'c" assumes "module scale" shows "module.rep scale r = (\<Sum>v\<in>keys r. scale (lookup r v) v)"
tmsubst0: fixes vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and t :: "tm"   and a :: "tm" shows "Itm vs (x # bs) (tmsubst0 t a) = Itm vs (Itm vs (x # bs) t # bs) a"
not_prefix_op2_next: fixes x :: "'r"   and xs :: "[:'r:]"   and pat :: "[:'r:]"   and usvs :: "[:'r:] \<times> [:'r:]" assumes "prefix\<cdot>[:x:]\<cdot>xs = FF" shows "op2\<cdot>pat\<cdot>(next\<cdot>xs\<cdot>(rep2\<cdot>pat\<cdot>usvs))\<cdot>x = op2\<cdot>pat\<cdot>(rep2\<cdot>pat\<cdot>usvs)\<cdot>x"
cop2_os__osp: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and X :: "'p"   and D :: "'p"   and E :: "'p"   and Y :: "'p" assumes "\<not> Coplanar A B C X"   and "Coplanar A B C D"   and "Coplanar A B C E"   and "D E OS X Y" shows "A B C OSP X Y"
M_M: fixes x :: "int" shows "pl.M (pl.M x) = pl.M x"
scalar_prodI_def: fixes ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and v :: "'c list"   and w :: "'c list" shows "scalar_prodI ze pl ti v w = foldr (\<lambda>(x, y). pl (ti x y)) (zip v w) ze"
trans_greater: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b = Greater"   and "cmp b c = Greater" shows "cmp a c = Greater"
actionCases: fixes \<alpha> :: "'d action" obtains M :: "'d" and N :: "'d" where "\<alpha> = Ma__\<lparr>Na__\<rparr>"   | M :: "'d" and xvec :: "name list" and N :: "'d" where "\<alpha> = M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>"   | "\<alpha> = \<tau>"
iNext_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNext t0 I P = (\<box> t [\<dots>0] \<oplus> inext t0 I. P t)"
DERIV_g: fixes x :: "real" assumes "0 < x" shows "(g has_real_derivative - P x) (at x)"
eq: fixes x :: "real" shows "indicator {0..} (x / l) = indicator {0..} x"
Max_eq_iff: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and m :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "A \<noteq> {}" shows "(ordered_powerprod_lin.Max A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<preceq> m))"
staticSecretA_notin_parts_initStateB: fixes m :: "msg"   and A :: "agent"   and B :: "agent" assumes "m \<in> staticSecret A" shows "(m \<in> parts (initState B)) = (A = B)"
lam_simps_3: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : hfsetcat.prod a b \<rightarrow> c\<guillemotright>" shows "cod (hfsetcat.\<Lambda> a b c f) = hfsetcat.exp b c"
sup_right_zero: fixes x :: "'a" shows "- x \<squnion> top = top"
closure_contains_Sup: fixes S :: "real set"   and B :: "real" assumes "S \<noteq> {}"   and "\<forall>x\<in>S. x \<le> B" shows "Sup S \<in> closure S"
div_field_poly_impl_def: fixes f :: "'b poly"   and g :: "'b poly" shows "div_field_poly_impl f g = (let cg = coeffs g in if cg = [] then 0 else let cf = coeffs f; ilc = inverse_class.inverse (last cg); ch = map ((*) ilc) cg; q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg)) in poly_of_list (map ((*) ilc) q))"
hRe_HComplex: fixes x :: "real star"   and y :: "real star" shows "hRe (HComplex x y) = x"
xy: shows "x = 2 * u * v \<and> y = u\<^sup>2 - v\<^sup>2 \<or> x = u\<^sup>2 - v\<^sup>2 \<and> y = 2 * u * v"
strategy_attracts_via_successor: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and A :: "'a set"   and W :: "'a set"   and w0 :: "'a" assumes "strategy p \<sigma>"   and "strategy_attracts_via p \<sigma> v0 A W"   and "v0 \<in> A - W"   and "v0 \<rightarrow> w0"   and "v0 \<in> VV p \<Longrightarrow> \<sigma> v0 = w0" shows "strategy_attracts_via p \<sigma> w0 A W"
simps_16: fixes x21 :: "char list"   and x22 :: "'a list" shows "set_literal (Neg x21 x22) = set x22"
setc_def_to_dest: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "A = {x. P x}"   and "x \<in> A" shows "P x"
residual_conj_eq_ext: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "residuated f" shows "residual (conjugate f) x = - f (- x)"
numsub_nz: fixes a :: "MIR.num"   and b :: "MIR.num" assumes "nozerocoeff a"   and "nozerocoeff b" shows "nozerocoeff (numsub a b)"
min_Liminf_at: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "min (f x) (Liminf (at x) f) = (SUP e\<in>{0<..}. Inf (f ` ball x e))"
assms: shows "R[x\<turnstile>n>y] = NotL <c>.N u"
not_invertible_iff_proj2_set_Col: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" shows "(\<not> invertible (vector [proj2_rep p, proj2_rep q, proj2_rep r])) = proj2_set_Col {p, q, r}"
Class_def: shows "e2.Class = (\<lambda>a\<in>Sigma I \<FF>2. {b \<in> Sigma I \<FF>2. (b, a) \<in> {(x, y). stk2.rel x y}})"
restrict_map_disj_dom: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h :: "'c \<Rightarrow> 'e option" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "h |` dom h\<^sub>0 \<bottom> h |` dom h\<^sub>1"
elims: fixes x :: "'b"   and xa :: "'b heap"   and y :: "'b heap" assumes "insert x xa = y" obtains xb :: "'b" and h :: "'b heap" where "x = xb" and "xa = h" and "y = merge (Hp xb []) h"
m: fixes i :: "nat" shows "m i = mk (fst (r i)) (a i)"
sdiv_word_def: fixes v :: "'a word"   and w :: "'a word" shows "v sdiv w = word_of_int (sint v sdiv sint w)"
product_set_finite: fixes X :: "'b fset set set" assumes "\<And>x. x \<in> X \<Longrightarrow> finite x" shows "finite (\<Otimes> X)"
atLeastLessThan_iff: fixes i :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and l :: "'d"   and u :: "'d" shows "(i \<in> ord.atLeastLessThan less_eq less l u) = (less_eq l i \<and> less i u)"
prems: shows "fmdom' m\<^sub>1 \<inter> fmdom' (m\<^sub>2(k $$:= v)) = {}"
IDC_def: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and E :: "'e \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'f"   and d :: "'f"   and c :: "'a"   and H :: "('f \<times> 'e) set" shows "WHATWHERE.IDC SR E DA d c H \<equiv> (\<exists>m m'. WHATWHERE.d_equal DA d m m' \<and> \<not> WHATWHERE.d_equal DA d (WHATWHERE.NextMem SR c m) (WHATWHERE.NextMem SR c m')) \<and> (\<forall>m m'. WHATWHERE.dH_equal E DA d H m m' \<longrightarrow> WHATWHERE.d_equal DA d (WHATWHERE.NextMem SR c m) (WHATWHERE.NextMem SR c m'))"
\<T>_eq_\<Z>: shows "\<T> k (Suc u) = \<Z> k (Suc u)"
ipresIGFresh_def: fixes h :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'w, 'ae, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ai, 'aj, 'ad, 'x, 'ak, 'al) model_scheme" shows "ipresIGFresh h MOD MOD' \<equiv> \<forall>ys y s X. igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFresh MOD' ys y (h X)"
vec_tensor_prod: fixes A :: "'a tensor"   and B :: "'a tensor" shows "vec (local.tensor_prod_otimes A B) = concat (map (\<lambda>a. vec_smult a (vec B)) (vec A))"
Dirichlet_L_def: fixes m :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and s :: "complex" shows "Dirichlet_L m \<chi> s = (if s = 1 then if \<chi> = principal_dchar m then 0 else eval_fds (fds \<chi>) 1 else of_nat m powr - s * (\<Sum>k = 1..m. \<chi> k * hurwitz_zeta (real k / real m) s))"
assms_5: shows "t mod cycleLength (nC k) mem schedule (nC k)"
fold_pnorm_deriv: fixes w :: "'a list"   and s :: "'a rexp" shows "fold (\<lambda>a r. pnorm (deriv a r)) w s = flatten PLUS (fold pderiv_set w {s})"
Mem_Neg_sym: fixes x :: "tm"   and y :: "tm" shows "{x IN y, y IN x} \<turnstile> Fls"
t: shows "A * B = (B\<^sup>T * A\<^sup>T)\<^sup>T"
convergesI: fixes T :: "'b set set"   and x :: "'b"   and F :: "'b set set" assumes "nhds\<^bsub>T\<^esub> x \<subseteq> F" shows "F \<longlongrightarrow>\<^bsub>T\<^esub> x"
sharing_consis_share_all_until_volatile_write: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "length ts' = length ts"   and "\<forall>i<length ts. let (uu_, uu_, uu_, sb, uu_, \<O>, uu_) = ts ! i; (uu_, uu_, uu_, sb', uu_, \<O>', uu_) = ts' ! i in \<O>' = acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O> \<and> sb' = dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb" shows "sharing_consis (share_all_until_volatile_write ts \<S>) ts' \<and> dom (share_all_until_volatile_write ts \<S>) - dom \<S> \<subseteq> (\<Union>(uu_, uu_, uu_, uu_, uu_, \<O>, uu_)\<in>set ts. \<O>) \<and> dom \<S> - dom (share_all_until_volatile_write ts \<S>) \<subseteq> (\<Union>(uu_, uu_, uu_, sb, uu_, \<O>, uu_)\<in>set ts. all_acquired sb \<union> \<O>)"
C2: fixes y :: "name"   and \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and zvec :: "name list"   and S :: "('a, 'b, 'c) psi" assumes "y \<sharp> \<Psi>'"   and "y \<sharp> R"   and "zvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>*zvec\<rparr>R \<parallel> S), \<lparr>\<nu>*zvec\<rparr>R \<parallel> \<lparr>\<nu>y\<rparr>S) \<in> Rel"
simps_1: shows "ket_plus_k 0 = ket_plus"
p_decomp: shows "p = [:- c, 1::'f:] * q"
adv_neg: shows "Negligible.negligible (\<lambda>n. 3 * adv_OT12 n)"
subbicategory_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool" shows "subbicategory V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B Arr \<equiv> bicategory V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B \<and> subcategory V Arr \<and> subbicategory_axioms V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B Arr"
power_minus_even: fixes a :: "'b"   and n :: "nat" shows "(- a) ^ (2 * n) = a ^ (2 * n)"
K_finite_consistency: shows "consistency (mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>}))"
rsquarefree_roots: fixes p :: "'b poly" shows "rsquarefree p = (\<forall>a. \<not> (poly p a = (0::'b) \<and> poly (pderiv p) a = (0::'b)))"
hashcode_literal'_def: fixes s :: "String.literal" shows "hashcode_literal' s \<equiv> foldl (\<lambda>h x. h * 33 + uint32_of_int (of_char x)) 5381 (raw_explode s)"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "Right.arr f"   and "Right.cod f = b" shows "Right.comp b f = f"
f_nxt_empty': fixes f :: "'b \<Rightarrow> 'b ttree"   and x :: "'b"   and T :: "'b set" assumes "f x = TTree.empty" shows "f_nxt f T x = f"
epsclo_eq_Image: fixes M :: "'b nfa"   and Q :: "hf set" assumes "nfa M" shows "nfa.epsclo M Q = nfa.states M \<inter> (eps M)\<^sup>* `` Q"
aEval_aEvalUni: fixes var :: "nat"   and a :: "atom"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and a' :: "atomUni" assumes "convert_atom var a (xs' @ x # xs) = Some a'"   and "length xs' = var" shows "aEval a (xs' @ x # xs) = aEvalUni a' x"
proj_k_mat: fixes k :: "nat" assumes "k < K" shows "proj_k k = mat K K (\<lambda>(i, j). if i = j \<and> i = k then 1 else 0)"
oalist_inv_raw_sort_oalist_aux: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and ko :: "'a" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "ko.oalist_inv_raw (rep_key_order ko) (oalist_raw.sort_oalist_aux rep_key_order ko xs)"
rtranclp_into_tranclp1: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
prems_7: shows "reads_consistent True \<O>' m (r # sb)"
subterms_sterms_cterms: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp"   and pn :: "'a" assumes "wellformed \<Gamma>"   and "p \<in> subterms (\<Gamma> pn)" shows "sterms \<Gamma> p \<subseteq> cterms \<Gamma>"
next_sibling_pure: fixes new_child :: "(_) node_ptr"   and h :: "(_) heap" shows "pure (a_next_sibling new_child) h"
notEvContainsBothAtoms: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and p :: "predicate"   and vs :: "vbl list"   and j :: "nat" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "\<not> EV (contains f (i, FAtom Pos p vs)) \<or> \<not> EV (contains f (j, FAtom Neg p vs))"
induct: fixes x :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "coding_hf x"   and "\<And>x. \<exists>i. x = ord_of i \<Longrightarrow> P x"   and "\<And>x y. \<lbrakk>coding_hf x; P x; coding_hf y; P y\<rbrakk> \<Longrightarrow> P \<langle>x, y\<rangle>" shows "P x"
intros_6: fixes n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"   and "n' \<in> sum_SDG_slice1 n" shows "n'' \<in> sum_SDG_slice1 n"
islimpt_iff_eventually: fixes x :: "'b"   and S :: "'b set" shows "x islimpt S = (\<not> (\<forall>\<^sub>F y in at x. y \<notin> S))"
shows_prec_artic_append: fixes p :: "nat"   and a :: "arctic"   and r :: "char list"   and s :: "char list" shows "shows_prec p a (r @ s) = shows_prec p a r @ s"
SMP_I': fixes N :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s N"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"   and "has_all_wt_instances_of \<Gamma> N M"   and "t \<in> N" shows "t \<in> SMP M"
abs_supp_2: fixes a :: "NominalSequents.var"   and x :: "'a" shows "supp ([a].x) = supp x - {a}"
dghm_dag_Rel_is_iso_dghm: shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha> : op_dg (dg_Rel \<alpha>) \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> dg_Rel \<alpha>"
tendsto_Limsup: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" assumes "F \<noteq> bot"   and "Limsup F f = Liminf F f" shows "(f \<longlongrightarrow> Limsup F f) F"
left_unit: fixes a :: "'a" assumes "a \<in> A" shows "addA zeroA a = a"
jipsen3l: fixes x :: "'a" shows "(1::'a) \<le> x \<leftarrow> x"