hcju/isabelleps huggingface dataset:
union_inter_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite A"   and "finite B"   and "\<forall>x\<in>A \<inter> B. g x = z" shows "abel_semigroup_set.F f z g (A \<union> B) = f (abel_semigroup_set.F f z g A) (abel_semigroup_set.F f z g B)"
assms: shows "size (A \<times># B) = size A * size B"
owner_seskey_unique: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Owner m, Crypt (SesK SK) (Pwd m)) \<in> s"   and "(Owner n, Crypt (SesK SK) (Pwd n)) \<in> s" shows "m = n"
smallomega_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "\<omega>[F](g) = {f. \<forall>c>0. \<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)}"
span_breakdown_eq: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and a :: "'c"   and S :: "'c set" assumes "module scale" shows "(x \<in> module.span scale (insert a S)) = (\<exists>k. x - scale k a \<in> module.span scale S)"
coprime_mod_imp_coprime: fixes p :: "int"   and f :: "int poly"   and g :: "int poly" assumes "prime p"   and "poly_mod.coprime_m p f g"   and "comm_monoid_mult_class.coprime (lead_coeff f) p \<or> comm_monoid_mult_class.coprime (lead_coeff g) p"   and "content f = 1 \<or> content g = 1" shows "comm_monoid_mult_class.coprime f g"
prWlsAbs_def: fixes gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'h \<Rightarrow> 'i) \<Rightarrow> ('varSort, 'i) sAbs"   and SEM :: "('j, 'k, 'l, 'sort, 'm, 'i, 'n) semDom_scheme" shows "prWlsAbs gA SEM \<equiv> \<forall>us s A val. wlsAbs (us, s) A \<and> sWlsVal SEM val \<longrightarrow> sWlsAbs SEM (us, s) (gA A val)"
elims_3: fixes x :: "'a Point"   and xa :: "'a Cone" assumes "\<not> onCone x xa" obtains p :: "'a Point" and cone :: "'a Cone" where "x = p" and "xa = cone" and "space2 (vertex cone) p \<noteq> slope cone * slope cone * time2 (vertex cone) p"
dv: shows "d * v = A $ from_nat i $ from_nat i"
assms_3: shows "(r, Store e i r) \<in> set (node_deliver_messages xs)"
negGorbit_list_nth: fixes i :: "nat"   and gs :: "'g list"   and T :: "'a \<Rightarrow> 'w"   and as :: "'a list" assumes "i < length gs" shows "HRep.negGorbit_list gs T as ! i = map ((*\<star>) (- gs ! i) \<circ> T) as"
vintersectionE: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A \<inter>\<^sub>\<circ> B" obtains "x \<in>\<^sub>\<circ> A" and "x \<in>\<^sub>\<circ> B"
norm: shows "\<forall>a\<in>set as. divisor a \<noteq> 0"
oseqp_sosp_oseqp_sos_eq: fixes \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" shows "oseqp_sosp \<Gamma> i = (\<lambda>x xa xb xc xd. ((x, xa), xb, xc, xd) \<in> oseqp_sos \<Gamma> i)"
set_ConsD: fixes y :: "'b"   and x :: "'b"   and xs :: "'b list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
\<alpha>': shows "LeftDerivation \<alpha> (take n' D) \<alpha>' \<and> ladder_\<gamma> (\<alpha> @ \<delta>) D L index = \<alpha>' @ \<delta> \<and> ladder_j L index < length \<alpha>'"
carrier_direct_lim_def: shows "dl.carrier_direct_lim \<equiv> eq.Partition"
interval_of_zero: shows "interval_of (0::'c) = 0"
dn: fixes x :: "'a vec" assumes "x \<in> set lst" shows "dim_vec x = n"
subst_TVar_scheme: fixes sch :: "type_scheme" shows "$ TVar sch = sch"
diagonal_to_Smith_PQ_def: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "diagonal_to_Smith_PQ A bezout = diagonal_to_Smith_aux_PQ [0..<min (nrows A) (ncols A) - 1] bezout (mat (1::'b), A, mat (1::'b))"
mult_B_mono: fixes w :: "(real, 'n) Finite_Cartesian_Product.vec"   and v :: "(real, 'n) Finite_Cartesian_Product.vec"   and i :: "'n" assumes "\<And>i. w $h i \<le> v $h i" shows "(local.B *v w) $h i \<le> (local.B *v v) $h i"
simps_2: fixes l :: "dag"   and a :: "ref"   and r :: "dag" shows "DAG (Node l a r) = (a \<notin> set_of l \<and> a \<notin> set_of r \<and> DAG l \<and> DAG r)"
lift_valid_path: fixes m :: "'b"   and as :: "'a list"   and m' :: "'b" assumes "m -as\<rightarrow>\<^sub>\<surd>* m'"   and "m \<noteq> Entry \<or> m' \<noteq> Exit" shows "\<exists>es. CFGExit.valid_path' (Node m) es (Node m')"
prefixeq_iprefix_trans: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "nat \<Rightarrow> 'b" assumes "prefix xs ys"   and "ys \<sqsubseteq> f" shows "xs \<sqsubseteq> f"
assms: shows "(\<sigma>, SubnetS s\<^sub>1 s\<^sub>2) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) (\<lambda>_ _ _. True) (\<lambda>_ _. True)"
dist_discrete_def: fixes n :: "'a discrete"   and m :: "'a discrete" shows "dist n m = (if n = m then 0 else 1)"
assms: shows "fds_abs_converges (fds_deriv f / f) s"
t_2: shows "(v \<circ> r \<circ> s \<circ> t) \<longlonglongrightarrow> limsup (v \<circ> r \<circ> s)"
eventually_frequently_simps_11: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "(\<not> frequently P F) = (\<forall>\<^sub>F x in F. \<not> P x)"
assms_3: shows "nsqn rt1 i = nsqn rt2 i"
\<pi>1l: shows "\<pi>\<^sub>1 l = \<pi> l"
ownership_distinct: 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 i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'b"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'b memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'b"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'b memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "ownership_distinct ts"   and "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {}"
le_neq_trans: fixes a :: "'b list"   and b :: "'b list" assumes "prefix a b"   and "a \<noteq> b" shows "strict_prefix a b"
has_vector_derivative_real_power: fixes x :: "real"   and c :: "'a"   and A :: "real set" assumes "0 < x" shows "((\<lambda>y. real_power y c) has_vector_derivative c * real_power x (c - (1::'a))) (at x within A)"
Hermite_of_row_i_works2: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'c" assumes "\<not> is_zero_row i A" shows "Hermite_of_row_i ass res A i $ i $ b = mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) $ i $ b"
ord_of_inject: fixes i :: "nat"   and j :: "nat" shows "(ord_of i = ord_of j) = (i = j)"
assms: shows "F \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)\<lparr>Arr\<rparr>"
sum_mset_image_mset_sum_mset_image_mset: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b multiset"   and A :: "'c multiset" shows "\<Sum>\<^sub># (image_mset g (\<Sum>\<^sub># (image_mset f A))) = (\<Sum>x\<in>#A. \<Sum>\<^sub># (image_mset g (f x)))"
imbal_del_Some: fixes x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "del x t = Some t'" shows "real (imbal t') - real (imbal t) \<le> 1"
the_elem_eq: fixes x :: "'e" shows "the_elem {x} = x"
resets_mono': fixes xs :: "('c \<Rightarrow> real) stream"   and i :: "nat"   and j :: "nat"   and x :: "'c"   and c :: "'c" assumes "PTA_Reachability.pairwise eq_elapsed xs"   and "pred_stream (\<lambda>u. u \<in> V) xs"   and "stream_trans xs"   and "\<forall>k\<ge>i. k \<le> j \<longrightarrow> zero x ([xs !! k]\<^sub>\<R>)"   and "x \<in> \<X>"   and "c \<in> \<X>"   and "i \<le> j" shows "(xs !! j) c = (xs !! i) c \<or> (xs !! j) c = 0"
AnalBinds_delete_lookup: fixes exp :: "exp \<Rightarrow> 'd \<rightarrow> 'c"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'd\<^sub>\<bottom>" shows "(ExpAnalysis.AnalBinds exp (delete x \<Gamma>)\<cdot>ae) x = \<bottom>"
simps_4: fixes f :: "'d + 'f"   and args :: "'e \<Rightarrow> ('d + 'f, 'e) trm"   and \<sigma> :: "'f \<Rightarrow> ('d, 'e) trm" shows "TsubstFO ($f f args) \<sigma> = (case f of Inl f' \<Rightarrow> $f f' (\<lambda>i. TsubstFO (args i) \<sigma>) | Inr f' \<Rightarrow> \<sigma> f')"
sprem_subs_n: shows "set (side_prems_of \<iota>) \<subseteq> N"
apex_def: shows "SPN_rs.cod.apex \<equiv> Maps.dom SPN_rs.cod.leg0"
finished_item_\<alpha>: shows "finished_item \<alpha> \<in> \<II>"
freshPerm: fixes x :: "name"   and p :: "(name \<times> name) list" assumes "x \<sharp> p" shows "p \<bullet> x = x"
filter_def: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b odlist" shows "ODList.filter P xs \<equiv> ODList (List.filter P (toList xs))"
word_must_wrap: fixes x :: "'b word"   and n :: "'b word" assumes "x \<le> n - 1"   and "n \<le> x" shows "n = 0"
assms_1: shows "f \<in> borel_measurable (F 0)"
IH: shows "list_all2 (\<lambda>x1 x2. veq_structure x1 x2 \<and> (vno_abs x2 \<longrightarrow> x1 = x2)) ts us"
MGT: shows "\<forall>Z. \<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>c \<down> Normal s} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
shows_string_assoc: fixes x :: "char list"   and r :: "char list"   and s :: "char list" shows "shows_string x r @ s = shows_string x (r @ s)"
non_infI: fixes r :: "('b \<times> 'b) set" assumes "\<And>a f. (\<And>i. (f i, f (Suc i)) \<in> r) \<Longrightarrow> \<exists>i. (f i, a) \<notin> r" shows "non_inf r"
eq_\<I>_gpv_3: shows "\<forall>x. x \<in> results_gpv \<I> gpv \<longrightarrow> \<I> \<turnstile>g f x \<surd>"
set_disconnected_nodes_locs_impl: shows "set_disconnected_nodes_locs = local.a_set_disconnected_nodes_locs"
rel_Grp: fixes A1 :: "'e set"   and f1 :: "'e \<Rightarrow> 'g"   and A2 :: "'f set"   and f2 :: "'f \<Rightarrow> 'h" shows "rel_prod (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. Basic_BNFs.fsts x \<subseteq> A1 \<and> Basic_BNFs.snds x \<subseteq> A2} (map_prod f1 f2)"
diff_single_eq_union: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x \<in># M" shows "(M - (x, {#}) = N) = (M = x, N)"
union_eq_empty: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(M + N = {#}) = (M = {#} \<and> N = {#})"
tms_cong_ax_holds: fixes e :: "(name, nat) finfun" shows "eval_fmla e tms_cong_ax"
rangeC_def: fixes F :: "('b \<Rightarrow> 'a) set" shows "rangeC F = \<Union> (range ` F)"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>\<mu>'.map f : G' a \<rightarrow>\<^sub>D G b\<guillemotright>"
some_eta: shows "(\<lambda>t. \<chi>i. sol t $ i) = sol"
assms_2: shows "\<not> [:- a, 1::'a:] ^ Suc n dvd p"
Inr_in_Plus: fixes x :: "'b"   and A :: "'a set"   and B :: "'b set" shows "(Inr x \<in> A <+> B) = (x \<in> B)"
simps_1: fixes xs :: "('a \<times> bool) list" shows "prappend_signed_list xs [] = xs"
wlsFSw_recAll_unique_presCons: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsFSw MOD"   and "presCons h hA MOD" shows "(wls s X \<longrightarrow> h X = rec MOD X) \<and> (wlsAbs (us, s') A \<longrightarrow> hA A = recAbs MOD A)"
assms_1: shows "(z1, z2) = calc_ideal_point1_cvec A B"
com_sifum_secure_def: fixes c :: "'Com" shows "com_sifum_secure c = c \<sim>\<^bsub>mds\<^sub>s\<^esub> c"
matrix_inv_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "matrix_inv A = (SOME A'. A ** A' = mat (1::'d) \<and> A' ** A = mat (1::'d))"
pdevs_val_coll_strict: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list"   and f :: "nat \<Rightarrow> real" assumes "list_all (coll 0 x) xs"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"   and "x \<noteq> 0"   and "f \<in> UNIV \<rightarrow> {- 1<..<1}" obtains e :: "real" where "e \<in> {- 1<..<1}" and "pdevs_val f (pdevs_of_list xs) = e *\<^sub>R sum_list xs"
nprt_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.nprt zero inf x = inf x zero"
bounded_check_eqv_def: shows "bounded_check_eqv \<equiv> WS1S.bounded.check_eqv"
matrix_scaleR_right: fixes r :: "real"   and a :: "(('a, 'd) vec, 'c) vec"   and b :: "(('a, 'b) vec, 'd) vec" shows "r *\<^sub>R a ** b = r *\<^sub>R (a ** b)"
conjugation_is_surj: fixes g :: "'a"   and h :: "'a" assumes "g \<in> carrier G"   and "h \<in> carrier G" shows "g \<otimes> (inv g \<otimes> h \<otimes> g) \<otimes> inv g = h"
has_products_def: fixes I :: "'b set" shows "D.J.has_products I \<equiv> I \<noteq> UNIV \<and> (\<forall>Ja D. discrete_diagram Ja J D \<and> Collect (partial_magma.arr Ja) = I \<longrightarrow> (\<exists>a. D.J.has_as_product Ja D a))"
that: fixes i :: "'b" assumes "i \<in> A" shows "coprime (f i) a"
TS_a': fixes x :: "nat"   and y :: "nat"   and h :: "nat list"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs \<le> 2 * T\<^sub>p [x, y] qs (OPT2 qs [x, y]) \<and> TS_inv' (Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs) (last qs) [x, y] \<and> Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = 2"
sorted_a_le: fixes x :: "'a extended"   and l :: "nat"   and y :: "'a extended"   and u :: "nat" assumes "idx s x = Inr l"   and "idx s y = Inr u" shows "(l < u) = (a x < a y)"
C: fixes cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "P' x' \<Turnstile> cfs\<box>" shows "(case cfs ! 0 of (pc, s, stk) \<Rightarrow> \<lambda>(pc', s', stk'). pc' = pc + size (P' x') + I' s \<and> Q' s s' \<and> stk' = F' s stk) (cfs ! (length cfs - 1))"
span_subspace: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "module scale"   and "A \<subseteq> B"   and "B \<subseteq> module.span scale A"   and "module.subspace scale B" shows "module.span scale A = B"
negligible_times_polynomial: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "negligible f"   and "polynomial g" shows "negligible (\<lambda>x. f x * g x)"
simps_2: fixes \<sigma> :: "(char list \<times> 'c list) trace"   and v :: "'c list"   and i :: "nat"   and t1 :: "'c MFOTL.trm"   and t2 :: "'c MFOTL.trm" shows "MFOTL.sat \<sigma> v i (formula.Eq t1 t2) = (MFOTL.eval_trm v t1 = MFOTL.eval_trm v t2)"
left_idem: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "semilattice (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (f a (b::'b::type)) = f a b"
disjoint_elem: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<inter> B = {}"   and "x \<in> A" shows "x \<notin> B"
ExitBlockNormal: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and abrs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> ([], (nrms, abrs) # css, Normal s) \<rightarrow> (nrms, css, Normal s)"
assms_5: shows "a \<otimes> d = b \<otimes> c"
word_m1_ge: fixes y :: "'b word" shows "y \<le> word_pred 0"
weakSimStatEq: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<leadsto><Rel> Q"   and "eqvt Rel'"   and "\<Psi> \<simeq> \<Psi>'"   and "\<And>\<Psi>' R S \<Psi>''. \<lbrakk>(\<Psi>', R, S) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', R, S) \<in> Rel'" shows "\<Psi>' \<rhd> P \<leadsto><Rel'> Q"
degree_zero_imp_empty_roots: fixes p :: "'a list list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "degree p = 0" shows "UP.roots p = {#}"
blinfun_apply: fixes x :: "'c \<Rightarrow>\<^sub>L 'd" shows "blinfun_apply x \<in> {f. bounded_linear f}"
exhaust_sel: fixes option :: "'c option" obtains "option = None"   | "option = Some (the option)"
emeasure_bind_const': fixes M :: "'a measure"   and N :: "'b measure"   and X :: "'b set" assumes "subprob_space M"   and "subprob_space N" shows "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X * emeasure M (space M)"
solution_nsc_code_def: shows "solution_nsc_code = solution_nsc \<V>"
map_iterator_genord_rule_P: fixes it :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<times> 'g \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "'f \<Rightarrow> 'g option"   and R :: "'f \<times> 'g \<Rightarrow> 'f \<times> 'g \<Rightarrow> bool"   and I :: "'f set \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma>0 :: "'e"   and c :: "'e \<Rightarrow> bool"   and f :: "'f \<times> 'g \<Rightarrow> 'e \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> bool" assumes "map_iterator_genord it m R"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> it - {k} \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v'); \<forall>k' v'. k' \<notin> it \<and> m k' = Some v' \<longrightarrow> R (k', v') (k, v)\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<notin> it \<and> m k = Some v \<and> k' \<in> it \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v')\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
real_mon_conv_bound: fixes c :: "nat \<Rightarrow> real"   and x :: "real" assumes "\<And>n. c n \<le> c (Suc n)"   and "\<And>n. c n \<le> x" shows "\<exists>l. c\<up>l \<and> l \<le> x"
approx_star_of_HFinite: fixes x :: "'a star"   and D :: "'a" assumes "x \<approx> star_of D" shows "x \<in> HFinite"
B: shows "mC_C theta c c2 s s2 P (lift P2 F2)"
\<L>\<^sub>P_is_word: fixes a :: "symbol list" assumes "a \<in> \<L>\<^sub>P" shows "is_word a"
machin_pi: shows "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))"
resNil_2: fixes x :: "name"   and b :: "name"   and P :: "pi" shows "<\<nu>x>x{b}.P \<sim> \<zero>"
fds_nth_scale: fixes c :: "nat"   and f :: "'b fds"   and n :: "nat" shows "fds_nth (fds_scale c f) n = (if 0 < n \<and> is_nth_power c n then fds_nth f (nth_root_nat c n) else (0::'b))"
3: shows "\<forall>(k, \<gamma>)\<in>one_chain2 - ps2. line_integral_exists F basis \<gamma>"
asm: shows "\<nexists>m. Suc n = m\<^sup>2"
induct: fixes x :: "'b llist"   and P :: "'b llist \<Rightarrow> bool" assumes "lfinite x"   and "P LNil"   and "\<And>xs x. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (LCons x xs)" shows "P x"
ts_ord_alt: fixes x :: "'a"   and y :: "'a" shows "(n x \<sqsubseteq> n y) = (n x \<oplus> n y = n y)"
gcd_0_right: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd a zero = normalize a"
hyps_3: shows "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l P'"
unifier\<^sub>t\<^sub>s_def: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and ts :: "fterm set" shows "unifier\<^sub>t\<^sub>s \<sigma> ts = (\<exists>t'. \<forall>t\<in>ts. t \<cdot>\<^sub>t \<sigma> = t')"
x'_freshP: shows "(\<forall>Y\<in>terms. fresh xs x' Y) \<and> (\<forall>A\<in>abs. freshAbs xs x' A) \<and> (\<forall>rho\<in>envs. freshEnv xs x' rho)"
simps_6: fixes va :: "'b match_expr"   and vb :: "'b match_expr" shows "normalized_nnf_match (MatchNot (MatchAnd va vb)) = False"
simps_2: fixes v :: "'a" shows "Num_arc_delta v + MinInfty_delta = Num_arc_delta v"
E_wf_2: shows "snd ` flows_fix \<T> \<subseteq> hosts \<T>"
div2_plus_div2: fixes n :: "nat" shows "n div 2 + n div 2 = n - n mod 2"
subspace_def: fixes S :: "'b poly set" shows "poly_vs.subspace S = (0 \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. smult c x \<in> S))"
bounded_ball: fixes x :: "'b"   and e :: "real" shows "bounded (ball x e)"
codomains_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "codomains (g \<cdot> f) = codomains g"
countable_basis: fixes x :: "'c" obtains A :: "nat \<Rightarrow> 'c set" where "\<And>i. open (A i)" and "\<And>i. x \<in> A i" and "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> x"
replacement: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "small X" shows "small (f ` X)"
l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
Fax_satPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "TE.satPB \<xi> Fax"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "L.in_hom f a b" obtains "L.arr f" and "L.dom f = a" and "L.cod f = b"
rtranclp_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P a"   and "\<And>y z. \<lbrakk>r\<^sup>*\<^sup>* a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
apropsOf_L_stateOf: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and out :: "'a \<Rightarrow> 'b \<Rightarrow> 'd"   and AP' :: "('b, 'c, 'd) Noninterference.aprop set"   and \<pi> :: "(('a, 'b, 'c) state \<times> ('b, 'c, 'd) Noninterference.aprop set) stream" assumes "Kripke.wfp UNIV (GM_sec_model.\<delta> do) UNIV (GM_sec_model.L out) AP' \<pi>" shows "apropsOf \<pi> = GM_sec_model.L out (stateOf \<pi>)"