hcju/isabelleps huggingface dataset:
in_intra_SDG_path_in_slice2: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n'"   and "n'' \<in> set ns" shows "n'' \<in> sum_SDG_slice2 n'"
While_sound_aux: fixes pre :: "'b set"   and b :: "'b set"   and post :: "'b set"   and P :: "'b com"   and rely :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and x :: "('b com option \<times> 'b) list"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list" assumes "pre \<inter> - b \<subseteq> post"   and "\<Turnstile> P sat [pre \<inter> b, rely, guar, pre]"   and "\<And>s. (s, s) \<in> guar"   and "stable pre rely"   and "stable post rely"   and "x \<in> cptn_mod"   and "x = (Some (While b P), s) # xs"   and "x \<in> assum (pre, rely)" shows "x \<in> comm (guar, post)"
tendsto_eventually: fixes "open" :: "'e set \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and l :: "'e"   and net :: "'f filter" assumes "class.topological_space open"   and "\<forall>\<^sub>F x in net. f x = l" shows "filterlim f (topological_space.nhds open l) net"
assms_2: shows "integrable M (\<lambda>x. f (T x) - f x)"
non_neg_mat_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "non_neg_mat A \<equiv> \<forall>a\<in>elements_mat_h A. (0::'a) \<le> a"
in_quotient_imp_non_empty: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "equiv A r"   and "X \<in> A // r" shows "X \<noteq> {}"
sort_l_props_2: shows "set (sort_fun l) = S0"
Le_le_LeD: fixes a :: "'c"   and b :: "'c" assumes "Le a \<le> Le b" shows "a \<le> b"
select_convs_3: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "rt \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = rt"
D_def: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list" shows "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>' = ((r|\<sigma> @ \<tau>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r \<and> (r|\<tau> @ \<sigma>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r)"
power_float_intervalI: fixes x :: "real"   and X :: "float interval"   and n :: "nat"   and p :: "nat" assumes "x \<in>\<^sub>r X" shows "x ^ n \<in>\<^sub>r power_float_interval p n X"
not_less: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(\<not> x <s y) = (y \<le>s x)"
eq: shows "[b] \<^sup>@ ku \<cdot> Ref \<C> us = [b] \<^sup>@ kv \<cdot> Ref \<C> vs"
bind_eq_Some_conv: fixes f :: "'b option"   and g :: "'b \<Rightarrow> 'a option"   and x :: "'a" shows "(f \<bind> g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
connected_component_eq_empty: fixes S :: "'b set"   and x :: "'b" shows "(connected_component_set S x = {}) = (x \<notin> S)"
wellorder_Max_lemma: fixes k :: "'a"   and A :: "'a set" assumes "k \<in> A"   and "finite A" shows "Max A \<in> A \<and> k \<le> Max A"
subspace_diff: fixes Sa :: "'a set"   and x :: "'a"   and y :: "'a" assumes "Sa \<subseteq> S"   and "subspace Sa"   and "x \<in> Sa"   and "y \<in> Sa" shows "x - y \<in> Sa"
abc_list_crsp_def: fixes xs :: "nat list"   and ys :: "nat list" shows "abc_list_crsp xs ys = (\<exists>n. xs = ys @ 0 \<up> n \<or> ys = xs @ 0 \<up> n)"
lower: fixes P :: "'a \<Rightarrow> bool" assumes "eventually P F" shows "y \<le> Sup (f ` Collect P)"
ag_inc_zero: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "\<zero>\<^bsub>A\<^esub> \<in> carrier A"
nres_order_simps_3: fixes M :: "'a nres" shows "M \<le> FAIL"
interaction_bound_map_gpv': fixes h :: "'e \<Rightarrow> 'd"   and "consider" :: "'f \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'f"   and gpv :: "('h, 'i, 'd) gpv" assumes "surj h" shows "interaction_bound consider (map_gpv' f g h gpv) = interaction_bound (consider \<circ> g) gpv"
is_extensional: fixes f :: "'a" assumes "\<not> C.arr f" shows "trg\<^sub>C f = C.null"
assms_2: shows "k \<in> fst ` set (fst (list_of_oalistx xs)) \<union> fst ` set (fst (list_of_oalisty ys))"
cohomotopically_trivial_on_components: fixes S :: "'a set"   and T :: "'b set" assumes "locally connected S \<or> compact S"   and "ANR T" shows "(\<forall>f g. continuous_on S f \<longrightarrow> f ` S \<subseteq> T \<longrightarrow> continuous_on S g \<longrightarrow> g ` S \<subseteq> T \<longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g) = (\<forall>C\<in>components S. \<forall>f g. continuous_on C f \<longrightarrow> f ` C \<subseteq> T \<longrightarrow> continuous_on C g \<longrightarrow> g ` C \<subseteq> T \<longrightarrow> homotopic_with_canon (\<lambda>x. True) C T f g)"
inext_nth_card_append_eq2: fixes A :: "nat set"   and B :: "nat set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "Max A < iMin B"   and "card A \<le> n" shows "(A \<union> B) \<rightarrow> n = B \<rightarrow> (n - card A)"
S_transform_weakly_bisimilar_iff: fixes P :: "'state"   and Q :: "'state" shows "weak_nominal_ts.weakly_bisimilar (\<turnstile>\<^sub>S) S_transform.S_transition (S_action.Act (S_action.Act \<tau>)) P Q = P \<approx>\<cdot>\<^sub>S Q"
right_diff_distrib': fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times a (minus b c) = minus (times a b) (times a c)"
simps_2: fixes k :: "'b" shows "atoms_of_lit (k\<inverse>) = k"
scaleR: fixes r :: "real"   and b :: "'a" shows "(if x \<in> S then f' x else (\<lambda>x. 0::'b)) (r *\<^sub>R b) = r *\<^sub>R (if x \<in> S then f' x else (\<lambda>x. 0::'b)) b"
fresh_3: fixes a :: "name"   and x1 :: "pi" shows "a \<sharp> \<tau>.x1 = a \<sharp> x1"
intro: fixes R :: "('e, 'f) ring_scheme" assumes "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<And>a b. \<lbrakk>a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>; a \<in> carrier R; b \<in> carrier R\<rbrakk> \<Longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>" shows "domain_axioms R"
set: fixes f :: "nat \<Rightarrow> 'b" shows "sum f {..<2} = f 0 + f 1"
PivotUpdateMinVars_axioms: shows "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"
extended_Gromov_product_boundary_inside_aux: fixes y :: "'a Gromov_completion"   and e :: "'a"   and x :: "'a" assumes "y \<in> Gromov_boundary" shows "extended_Gromov_product_at e y (to_Gromov_completion x) = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e (v n) x)) |v. (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
pre_zeta_aux_eq_pre_zeta: fixes n :: "nat"   and s :: "complex"   and a :: "real" assumes "- 2 * real n < Re s"   and "0 < a" shows "pre_zeta_aux n a s = pre_zeta a s"
opI: fixes a :: "'a" assumes "a \<in> S" shows "\<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. R a x)"
nrename_AndL1: fixes R :: "trm"   and x :: "name"   and y :: "name"   and z :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = AndL1 z.N u"   and "z \<sharp> (R, x, y)"   and "u \<sharp> (x, y)" shows "\<exists>N'. R = AndL1 z.N' u \<and> N'[x\<turnstile>n>y] = N"
balance_tree_def: fixes t :: "'b tree" shows "balance_tree t = Balance.bal_tree (size t) t"
eq_onp_mono0: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>A. P x \<longrightarrow> Q x" shows "\<forall>x\<in>A. \<forall>y\<in>A. eq_onp P x y \<longrightarrow> eq_onp Q x y"
Keys_mono: fixes A :: "('e \<Rightarrow>\<^sub>0 'f) set"   and B :: "('e \<Rightarrow>\<^sub>0 'f) set" assumes "A \<subseteq> B" shows "Keys A \<subseteq> Keys B"
id_rad_set: fixes I :: "'a set" assumes "ideal R I" shows "rad_ideal R I = {x \<in> carrier R. \<exists>n. x^\<^bsup>R n\<^esup> \<in> I}"
assms_2: fixes \<sigma> :: "nat \<Rightarrow> ('a, 'b, 'c, 'd, 'e) system_state" assumes "prerun sys \<sigma>" shows "(\<box>(\<lambda>s. \<lceil>I\<rceil> s \<longrightarrow> (\<circle>\<lceil>I\<rceil>) s)) \<sigma>"
finite_measure_eq_AE: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b pmf" assumes "AE x in measure_pmf M. (x \<in> A) = (x \<in> B)"   and "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M A = measure_pmf.prob M B"
InterpVar: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a"   and c :: "'c" assumes "L\<lbrakk>\<turnstile> A Type\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IObj c" shows "L\<lbrakk>Vx : A \<turnstile> Vx : A\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor (id\<^bsub>iC\<^bsub>I\<^esub>\<^esub> c)"
Inf_nres_def: fixes X :: "'a nres set" shows "Inf X \<equiv> if \<exists>x. RES x \<in> X then RES (\<Inter> {x. RES x \<in> X}) else FAILi"
SPRAY_event: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.SPRAY \<P> x \<noteq> {}" shows "x \<in> \<E>"
inti: fixes i :: "'b" shows "LINT x|M. indicat_real A (T x) * f i x = LINT x|M. indicat_real A x * real_transfer_operator (f i) x"
close_def: fixes C :: "('a, 'b) form set set" shows "close C = {S. \<exists>S'\<in>C. S \<subseteq> S'}"
that: fixes h' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> ok g x" shows "thesis"
valid_plan_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list" shows "ast_problem.valid_plan problem \<pi>s \<equiv> \<exists>s'\<in>ast_problem.G problem. ast_problem.path_to problem (ast_problem.I problem) \<pi>s s'"
dg_Dom_vdomain: shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
not_eq_last_diff_paths_i: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and x :: "('a, 'b) fmap list" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "x \<in> state_set (state_list s as)" shows "last x \<in> valid_states PROB"
sub_designI: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and \<U> :: "'b set"   and \<A> :: "'b set multiset" assumes "design point_set block_collection"   and "design \<U> \<A>"   and "sub_incidence_system \<U> \<A> point_set block_collection" shows "sub_design \<U> \<A> point_set block_collection"
lincomb_index: fixes i :: "nat"   and n :: "nat"   and X :: "'b vec set"   and a :: "'b vec \<Rightarrow> 'b" assumes "i < n"   and "X \<subseteq> carrier_vec n" shows "module.lincomb (module_vec TYPE('b) n) a X $ i = (\<Sum>x\<in>X. a x * x $ i)"
Adjunct_segment_sub: fixes X :: "'a set" assumes "S_inductive_set D"   and "Chain D X" shows "segment (Iod D (insert (Algebra2.Sup D X) X)) (Algebra2.Sup D X) \<subseteq> X"
expand: fixes mc :: "mc"   and mc' :: "mc" assumes "un_MC mc = un_MC mc'" shows "mc = mc'"
simps_2: shows "(dFAILi \<le> dFAILi) = True"
liftprj_eq: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and liftemb :: "'e\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'e\<^sub>\<bottom>"   and liftdefl :: "'e itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'e \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'e"   and defl :: "'e itself \<Rightarrow> udom defl" assumes "class.domain below liftemb liftprj liftdefl emb prj defl" shows "liftprj = u_map\<cdot>prj"
mult_sub_right_one: fixes x :: "'a" shows "x \<le> x * (1::'a)"
trans: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "blinding_of_on A h bo"   and "bo x y"   and "bo y z"   and "x \<in> A" shows "bo x z"
gval_Lt: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "gval (Lt a1 a2) s = value_gt (aval a2 s) (aval a1 s)"
ide_leg1: shows "ide fg\<^sub>0h\<^sub>1.p\<^sub>1"
CallRedsNull: fixes e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>null,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s\<^sub>2,b\<^sub>2\<rangle>"
simps: fixes a :: "'d \<Rightarrow> real"   and X :: "'d set"   and I :: "'d \<Rightarrow> intv"   and r :: "('d \<times> 'd) set" shows "(a \<in> region X I r) = (\<exists>u X\<^sub>0. a = u \<and> (\<forall>x\<in>X. 0 \<le> u x) \<and> (\<forall>x\<in>X. intv_elem x u (I x)) \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d} \<and> (\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))))"
adjoint_pair_preserved_by_iso: fixes f :: "'a"   and g :: "'a"   and \<phi> :: "'a"   and f' :: "'a"   and \<psi> :: "'a"   and g' :: "'a" assumes "adjoint_pair f g"   and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>"   and "local.iso \<phi>"   and "\<guillemotleft>\<psi> : g \<Rightarrow> g'\<guillemotright>"   and "local.iso \<psi>" shows "adjoint_pair f' g'"
intF: fixes f :: "('fsym, 'tp) efsym"   and al :: "univ list" assumes "TE_wtFsym f"   and "list_all2 eintT (TE_arOf f) al" shows "eintT (TE_resOf f) (eintF f al)"
map_by_foldl: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list" shows "foldl (\<lambda>l x. l @ [f x]) [] l = map f l"
wtC_def: shows "TE.wtC \<equiv> list_all TE.wtL"
nu0: shows "- s \<le> real_of_int n * u - s"
matchC_LC_mono: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and theta' :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set" assumes "PL_Indis cval indis"   and "PL_Indis.matchC_LC aval tval cval indis theta c d"   and "theta \<subseteq> theta'" shows "PL_Indis.matchC_LC aval tval cval indis theta' c d"
bind_return: fixes m :: "'a + 'b" shows "m \<bind> Inr = m"
assms: shows "(c', ictxt, fctxt) \<in> decompose_com c"
cases: fixes x :: "('d \<times> 'e \<times> 'f) tree" obtains uu_ :: "('d \<times> 'e \<times> 'f) tree" and uv_ :: "'d" and uw_ :: "'e" and a :: "'f" and ux_ :: "('d \<times> 'e \<times> 'f) tree" where "x = \<langle>uu_, (uv_, uw_, a), ux_\<rangle>"   | "x = \<langle>\<rangle>"
exec_wfs_always_match: fixes \<Gamma> :: "char list set"   and s :: "stmt"   and \<sigma> :: "char list \<Rightarrow> nat option" assumes "wfs \<Gamma> s"   and "match \<sigma> \<Gamma>" shows "\<exists>\<sigma>'. s \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<and> match \<sigma>' \<Gamma>"
Pred: fixes f :: "'effect"   and F :: "'effect fs_set"   and \<phi> :: "'pred" assumes "f \<in>\<^sub>f\<^sub>s F" shows "Pred f \<phi> \<in> \<A>[F]"
elims_2: fixes x :: "'d \<times> 'c" assumes "valid_out_port x" obtains v :: "'d" and p :: "'c" where "x = (v, p)" and "v |\<in>| vertices \<and> p |\<in>| outPorts (nodeOf v)"
scopeExtSumLeft: fixes x :: "name"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" assumes "x \<sharp> P"   and "\<And>y R. y \<sharp> R \<Longrightarrow> (\<lparr>\<nu>y\<rparr>R, R) \<in> Rel"   and "Id \<subseteq> Rel" shows "\<lparr>\<nu>x\<rparr>(P \<oplus> Q) \<leadsto>[Rel] P \<oplus> \<lparr>\<nu>x\<rparr>Q"
bounded: fixes f :: "'b \<Rightarrow> real" assumes "f \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x \<le> B" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>join M \<noteq> top"
outSetCorelatedNonemptyX: fixes x :: "chanID" assumes "OUTfromV x \<noteq> {}"   and "OUTfromV_VARto" shows "x \<in> outSetCorelated x"
\<delta>: shows "is_derivation (terminals p @ item_\<beta> x @ \<delta>)"
connectedin_continuous_map_image: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "continuous_map X Y f"   and "connectedin X S" shows "connectedin Y (f ` S)"
H_1xn_Q3: shows "(H_1xn, Q3) = Smith_1xn H2_DR"
funpow_mod_eq: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and m :: "nat" assumes "(f ^^ n) x = x"   and "0 < n" shows "(f ^^ (m mod n)) x = (f ^^ m) x"
le_infI1: fixes a :: "'k multiset"   and x :: "'k multiset"   and b :: "'k multiset" assumes "a \<subseteq># x" shows "a \<inter># b \<subseteq># x"
l17: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) = - (x \<sqinter> - - y)"
forks_update_take: fixes n :: "nat"   and fs :: "nat list"   and e :: "dining_event" shows "take n (Up\<^sub>F fs e) = Up\<^sub>F (take n fs) e"
steps_hd_drop_suffix_infinite: fixes f :: "nat \<Rightarrow> ('s, 'p, 'f) com list \<times> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list \<times> ('s, 'f) xstate"   and c :: "('s, 'p, 'f) com"   and cs :: "('s, 'p, 'f) com list"   and css :: "(('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and s :: "('s, 'f) xstate"   and \<Gamma> :: "'p \<Rightarrow> ('s, 'p, 'f) com option"   and pcss :: "nat \<Rightarrow> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and pcs :: "nat \<Rightarrow> ('s, 'p, 'f) com list"   and i :: "nat" assumes "f 0 = (c # cs, css, s)"   and "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"   and "\<forall>i. \<not> (CS (f i) = cs \<and> CSS (f i) = css)"   and "\<forall>i. if pcss i = [] then CSS (f i) = css \<and> CS (f i) = pcs i @ cs else CS (f i) = pcs i \<and> CSS (f i) = butlast (pcss i) @ [(fst (last (pcss i)) @ cs, snd (last (pcss i)) @ cs)] @ css" shows "\<Gamma>\<turnstile> (pcs i, pcss i, S (f i)) \<rightarrow> (pcs (Suc i), pcss (Suc i), S (f (Suc i)))"
3: shows "nth_least s ` liset A (lselect s w) = s \<inter> liset A w"
assms_6: shows "diff_inv = (\<pi> - initial_value pf) / constant_image (prices Mkt pos_stock 0)"
abs_conv_abscissa_diff_leI: fixes f :: "'b fds"   and d :: "ereal"   and g :: "'b fds" assumes "abs_conv_abscissa f \<le> d"   and "abs_conv_abscissa g \<le> d" shows "abs_conv_abscissa (f - g) \<le> d"
simps: fixes u :: "'a Vector" shows "~ u = \<lparr>tdir = - tdir u, xdir = - xdir u, ydir = - ydir u, zdir = - zdir u\<rparr>"
rembiflowdups_complete: fixes x :: "('a \<times> 'a) list" assumes "\<forall>(s, r)\<in>set x. (r, s) \<in> set x" shows "set (rembiflowdups x) \<union> set (backlinks (rembiflowdups x)) = set x"
indRelRSTPO_impl_SRel_and_TRel_weakly_preserve_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelRSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs TRel TWB"
SeqWRP_cong: fixes H :: "fm set"   and s :: "tm"   and s' :: "tm"   and k :: "tm"   and k' :: "tm"   and y :: "tm"   and y' :: "tm" assumes "H \<turnstile> s EQ s'"   and "H \<turnstile> k EQ k'"   and "H \<turnstile> y EQ y'" shows "H \<turnstile> SeqWRP s k y IFF SeqWRP s' k' y'"
ld: shows "LeftDerivationLadder \<alpha> D L \<gamma>"
type_copy_map_cong0: fixes M :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c"   and N :: "'d \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'd"   and f :: "'a \<Rightarrow> 'e" assumes "M (g x) = N (h x)" shows "(f \<circ> M \<circ> g) x = (f \<circ> N \<circ> h) x"
length_sort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "length (linorder.sort_key less_eq f xs) = length xs"
w: fixes n :: "nat" assumes "WtFT cl \<noteq> 1 \<and> n < length cl \<and> \<not> finished (cl ! n)" shows "w n = 1 / real (length cl) / (1 - WtFT cl)"
G_def: shows "G = \<lparr>edge = \<lambda>x y. 0 < cap' (x, y)\<rparr>"
simps_14: fixes x1 :: "'b"   and x41 :: "'b local.term"   and x42 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> x41 \<^bold>\<cdot> x42"
one_not_Infinitesimal: shows "1 \<notin> Infinitesimal"
hyps_3: fixes thesis :: "bool" assumes "\<And>W. \<lbrakk>W \<in> I; \<forall>a\<in>F. W \<subseteq> a\<rbrakk> \<Longrightarrow> thesis"   and "F \<noteq> {}"   and "F \<subseteq> I" shows "thesis"
generator_Nil_prod: shows "generator Nil_prod = (\<lambda>_. Done)"
walk_in_V: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list" assumes "Graph G"   and "Graph.walk G xs" shows "set xs \<subseteq> V\<^bsub>G\<^esub>"
10: shows "prim_EP w v e \<le> top * e * prim_P w v e\<^sup>T\<^sup>\<star>"
unrest_uminus: fixes x :: "'a \<Longrightarrow> 'b"   and u :: "('c, 'b) uexpr" assumes "x \<sharp> u" shows "x \<sharp> - u"
posct: shows "\<forall>k<x1. positive ((map denote x3a ! k) (x2a k * x4 * adjoint (x2a k))) \<and> (map denote x3a ! k) (x2a k * x4 * adjoint (x2a k)) \<in> carrier_mat d d \<and> trace ((map denote x3a ! k) (x2a k * x4 * adjoint (x2a k))) \<le> trace (x2a k * x4 * adjoint (x2a k))"
AUX: fixes S :: "(nat \<times> nat) set" assumes "finite S" shows "length (filter (\<lambda>a. \<exists>e\<in>S. a = SAT_PUSH' e) p) = (\<Sum>e\<in>S. length (filter ((=) (SAT_PUSH' e)) p))"
stock_price_borel_measurable: shows "borel_adapt_stoch_proc G (prices Mkt stk)"
long_division_add_iff: fixes K :: "'a list set"   and a :: "'a list list"   and b :: "'a list list"   and c :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "a \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "b \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "c \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "(UP.pmod a q = UP.pmod b q) = (UP.pmod (a \<oplus>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> c) q = UP.pmod (b \<oplus>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> c) q)"
inner: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> restricted_subformulas_inner \<phi>" shows "size \<psi> \<le> size \<phi>"
assms_6: fixes i :: "nat"   and i' :: "nat" assumes "i < nr"   and "f i < nc"   and "i' < nr"   and "i' \<noteq> i" shows "A $$ (i', f i) = (0::'a)"
Re_plus_Re: fixes x :: "'a"   and y :: "'a" shows "Re (cinner x y) + Re (cinner y x) = 2 * Re (cinner x y)"
low_equal_def: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" shows "low_equal s1 s2 \<equiv> cpu_reg s1 = cpu_reg s2 \<and> user_reg s1 = user_reg s2 \<and> sys_reg s1 = sys_reg s2 \<and> (\<forall>va. virt_to_phys va (mmu s1) (mem s1) = virt_to_phys va (mmu s2) (mem s2)) \<and> (\<forall>pa. user_accessible s1 pa \<longrightarrow> mem_equal s1 s2 pa) \<and> mmu s1 = mmu s2 \<and> state_var s1 = state_var s2 \<and> traps s1 = traps s2 \<and> undef s1 = undef s2"