hcju/isabelleps huggingface dataset:
image_ring_is_ring: fixes f :: "('c multiset \<Rightarrow> 'a) list \<Rightarrow> 'd"   and I :: "('c multiset \<Rightarrow> 'a) list set" assumes "weak_ring_morphism f I (poly_ring L)" shows "ring (image_ring f (poly_ring L))"
IV2: shows "(\<not> check_eq r1 r2 T) = (\<exists>bs. bddh (length bs) r1 \<and> bddh (length bs) r2 \<and> (\<exists>n<m. dist_nodes M n v (bdd_lookup r1 bs) (bdd_lookup r2 bs)))"
assms_5: shows "path_image \<gamma> \<subseteq> S"
homeomorphism_apply2: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "homeomorphism S T f g"   and "x \<in> T" shows "f (g x) = x"
multiple_mask_trivia: fixes m :: "nat"   and n :: "nat"   and x :: "'a word" assumes "m \<le> n" shows "(x AND NOT (mask n)) + (x AND mask n AND NOT (mask m)) = x AND NOT (mask m)"
bigo_lesseq4: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(h)"   and "\<forall>x. (0::'b) \<le> g x"   and "\<forall>x. g x \<le> \<bar>f x\<bar>" shows "g \<in> O(h)"
reversepath_def: fixes g :: "real \<Rightarrow> 'b" shows "reversepath g = (\<lambda>x. g (1 - x))"
fun_evaluate_length_1: fixes s :: "'a state"   and env :: "v sem_env"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and s' :: "'a state"   and res :: "(v list, v) result" assumes "fun_evaluate_match s env v pes err_v = (s', res)" shows "case res of Rval vs \<Rightarrow> length vs = 1 | Rerr error_result \<Rightarrow> True"
one_strongly_connected_iff_6_aux: fixes x :: "'a" shows "x ; x\<^sup>+ \<le> x ; 1 ; x"
b': shows "\<forall>r'\<in>R. rightPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* E) \<in> set (fst r')"
finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K"   and "finite (\<omega>_limit_set x)" obtains y :: "'a" where "\<omega>_limit_set x = {y}" and "f y = (0::'a)"
FrameStatEq_def: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and G :: "'d frame" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.FrameStatEq SImp F G \<equiv> assertionAux.FrameStatImp SImp F G \<and> assertionAux.FrameStatImp SImp G F"
dvd_add_right_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one"   and "dvd.dvd times a b" shows "dvd.dvd times a (plus b c) = dvd.dvd times a c"
cond_conj_distr: fixes P :: "'a upred"   and b :: "'a upred"   and Q :: "'a upred"   and S :: "'a upred" shows "(P \<and> Q \<triangleleft> b \<triangleright> S) = (P \<and> Q) \<triangleleft> b \<triangleright> (P \<and> S)"
prod: shows "poly_inv (monom_mult_poly (m, c) p)"
atLeastLessThan_concat: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and g :: "nat \<Rightarrow> 'd" assumes "m \<le> n"   and "n \<le> p" shows "sum g {m..<n} + sum g {n..<p} = sum g {m..<p}"
dependent_on_def: fixes s :: "'b set" shows "local.dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *s v) = (0::'b) \<and> (\<exists>v\<in>t. u v \<noteq> (0::'a)))"
obj_def: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.obj a \<equiv> Span.arr a \<and> Span.src a = a"
collapse: fixes dynamic_array :: "'a dynamic_array" shows "Dyn_Array (alen dynamic_array) (aref dynamic_array) = dynamic_array"
S0'_eq: shows "S0' = \<alpha> ` S0"
approx_remains_after_same_comp_left: fixes p :: "'a pmf"   and q :: "'a pmf"   and r :: "'a pmf"   and \<alpha> :: "real" assumes "p \<approx>[\<R>] q"   and "r \<in> local.\<P>"   and "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> r p \<approx>[\<R>] mix_pmf \<alpha> r q"
Cd_singleton: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" shows "(x \<in> Singleton_for_d.Cd d X) = (Singleton_for_d.Cd d X = {x})"
ctr: fixes x :: "quat"   and y :: "quat" shows "x * y = Quat (quat.Re x * quat.Re y - Im1 x * Im1 y - Im2 x * Im2 y - Im3 x * Im3 y) (quat.Re x * Im1 y + Im1 x * quat.Re y + Im2 x * Im3 y - Im3 x * Im2 y) (quat.Re x * Im2 y - Im1 x * Im3 y + Im2 x * quat.Re y + Im3 x * Im1 y) (quat.Re x * Im3 y + Im1 x * Im2 y - Im2 x * Im1 y + Im3 x * quat.Re y)"
sum_list_mult_const: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and c :: "'b"   and xs :: "'c list" assumes "class.semiring_0 plus zero times" shows "monoid_add.sum_list plus zero (map (\<lambda>x. times (f x) c) xs) = times (monoid_add.sum_list plus zero (map f xs)) c"
m_def: shows "m = (f l + f u - a * (l + u)) / 2"
infinite_descent: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m" shows "P n"
arrE: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "arr (fst f) \<and> arr (snd f) \<and> src (fst f) = trg (snd f)"
subst_iff: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and P :: "'b upred"   and Q :: "'b upred" shows "\<sigma> \<dagger> (P \<Leftrightarrow> Q) = (\<sigma> \<dagger> P \<Leftrightarrow> \<sigma> \<dagger> Q)"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "distinct (x # xs) = (x \<notin> list.set xs \<and> distinct xs)"
rep_eq: fixes x :: "complex mat" shows "(*\<^sub>V) (cblinfun_of_mat x) = (\<lambda>v. if x \<in> carrier_mat (length canonical_basis) (length canonical_basis) then basis_enum_of_vec (x *\<^sub>v vec_of_basis_enum v) else (0::'c))"
axioms_2: fixes R :: "('c, 'd) ring_scheme" assumes "semiring R" shows "Group.monoid R"
put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_put_ptrs: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('o, 'm, 'n, 'w option) RShadowRoot_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put shadow_root_ptr shadow_root h = h'" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast shadow_root_ptr|}"
hom_power: fixes x :: "int"   and n :: "nat" shows "of_int (x ^ n) = of_int x ^ n"
learn_lim_imp_BC: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_lim \<psi> U s" shows "learn_bc \<psi> U s"
append_eq_map_conv: fixes ys :: "'d list"   and zs :: "'d list"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "(ys @ zs = map f xs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
edges_ins_nodes: fixes v :: "'a"   and g :: "'a ugraph" shows "edges (ins_node v g) = edges g"
IH_1: shows "evalUni F1 x = evalUni (list_disj_Uni (map (list_conj_Uni \<circ> map AtomUni) (DNFUni F1))) x"
nvA_varsA: fixes x :: "Preliminaries.var"   and at :: "('c, 'd) atm" assumes "x \<in> nvA at" shows "x \<in> varsA at"
Fourier_products_integrable_cos: fixes f :: "real \<Rightarrow> real"   and k :: "real" assumes "f absolutely_integrable_on {- pi..pi}" shows "integrable (lebesgue_on {- pi..pi}) (\<lambda>x. cos (k * x) * f x)"
intros_3: fixes xs :: "'a list"   and ys :: "'a list"   and z :: "'a" assumes "xs <~~> ys" shows "z # xs <~~> z # ys"
jumpF_polyL_inverse_add: fixes p :: "real poly"   and q :: "real poly"   and x :: "real" assumes "coprime p q" shows "jumpF_polyL q p x + jumpF_polyL p q x = jumpF_polyL 1 (q * p) x"
non_strict_into_strict: fixes NS :: "('a \<times> 'a) set"   and S :: "('a \<times> 'b) set"   and s :: "'a"   and t :: "'b" assumes "NS O S \<subseteq> S"   and "(s, t) \<in> NS\<^sup>* O S" shows "(s, t) \<in> S"
map_simp_ide: fixes a :: "'c" assumes "Cop.ide a" shows "\<T>e.map a = \<T>o e a"
rep_eq: fixes x :: "'b sq_mtx" shows "(*\<^sub>V) x = (*v) (to_vec x)"
arg_rcis: fixes x :: "real"   and r :: "real" assumes "x \<in> {- pi<..pi}"   and "0 < r" shows "arg (rcis r x) = x"
n_eq_val_eq_idealTr: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and x :: "'a"   and y :: "'a" assumes "distinct_pds K n P"   and "x \<in> carrier (O\<^bsub>K P n\<^esub>)"   and "y \<in> carrier (O\<^bsub>K P n\<^esub>)"   and "\<forall>j\<le>n. (\<nu>\<^bsub>K P j\<^esub>) x \<le> (\<nu>\<^bsub>K P j\<^esub>) y" shows "(O\<^bsub>K P n\<^esub>) \<diamondsuit>\<^sub>p y \<subseteq> (O\<^bsub>K P n\<^esub>) \<diamondsuit>\<^sub>p x"
RestrictedP_Eats: fixes s :: "tm"   and k :: "tm"   and s' :: "tm"   and a :: "tm"   and b :: "tm" shows "{RestrictedP s k s', a IN k} \<turnstile> RestrictedP (Eats s (HPair a b)) k (Eats s' (HPair a b))"
left_inverse: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "Groups.group f z inverse" shows "f (inverse a) a = z"
simps_5: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "subc c s (Impl p q) = Impl (subc c s p) (subc c s q)"
MakeFaceFinal_faces_distinct: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "faces_distinct (makeFaceFinal f g)"
FINITE_snapshot: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" assumes "finite PROB" shows "finite (snapshot PROB s)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "\<And>c d qs. (\<not> c \<le> d \<Longrightarrow> P c (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))) \<Longrightarrow> P c d qs" shows "P a0 a1 a2"
power_pos: fixes a :: "'a"   and n :: "nat" assumes "(0::'a) < a" shows "(0::'a) < a ^ n"
measure_le_1: fixes M :: "'a measure"   and X :: "'a set" assumes "prob_space M" shows "emeasure M X \<le> 1"
l_set_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_child_nodes type_wf set_child_nodes set_child_nodes_locs \<equiv> (\<forall>ptr children h h'. writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h') \<and> (\<forall>w object_ptr h h'. w \<in> set_child_nodes_locs object_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h') \<and> (\<forall>w object_ptr h h'. w \<in> set_child_nodes_locs object_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h')"
ls: shows "list_all (\<lambda>i. i < length signs) (snd I)"
assms_1: shows "acyclicP_on (fmran' ym) P"
vector_to_cblinfun_code_def: shows "vector_to_cblinfun_code = vector_to_cblinfun"
neg_eq_simps_3: fixes x :: "tr" assumes "neg\<cdot>x = \<bottom>" shows "x = \<bottom>"
child_neighbour: fixes p :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat"   and ps :: "(nat \<times> int) list" assumes "child p (Grid_Point.inv dir) d = child ps dir d" shows "ps = p[d := (lv p d, ix p d - Grid_Point.sgn dir)]"
mcont_inf1: fixes x :: "'a" shows "mcont Sup (\<le>) Sup (\<le>) ((\<sqinter>) x)"
intros: fixes attributes :: "'a \<rightharpoonup>\<^sub>f 'b \<rightharpoonup>\<^sub>f 'c"   and \<C> :: "'a"   and attr :: "'b"   and \<D>' :: "'a"   and \<tau>' :: "'c"   and \<D> :: "'a"   and \<tau> :: "'c" assumes "closest_attribute attributes \<C> attr \<D>' \<tau>'"   and "\<D> \<noteq> \<D>' \<or> \<tau> \<noteq> \<tau>'" shows "closest_attribute_not_unique attributes \<C> attr \<D> \<tau>"
eq_diff0: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<ominus>\<^bsub>R\<^esub> y = \<zero>\<^bsub>R\<^esub>) = (x = y)"
add_neg_numeral_special_8: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) one = zero"
carrier_substitute_aux: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree"   and A :: "'b set" assumes "xs \<in> paths (substitute f T t)"   and "carrier t \<subseteq> A"   and "\<And>x. x \<in> A \<Longrightarrow> carrier (f x) \<subseteq> A" shows "set xs \<subseteq> A"
bisect_def: shows "bisect = (\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y))"
ideal_generated_UNIV_obtain_pair: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "ring.ideal_generated plus zero uminus times {a, b} = ring.ideal_generated plus zero uminus times {one}" shows "\<exists>p q. plus (times p a) (times q b) = d"
a_minus_minus: fixes z :: "ant" shows "- (- z) = z"
nonneg_cexpr_shift_iff: fixes V :: "nat set"   and t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" shows "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) (map_vars Suc e) = nonneg_cexpr V \<Gamma> e"
cases: fixes x :: "('a, 'b, 'c, 'd) Tree \<times> ('a, 'b, 'c, 'd) Tree" obtains tset1 :: "('a, 'b, 'c, 'd) Tree set['a]" and tset2 :: "('a, 'b, 'c, 'd) Tree set['a]" where "x = (tConj tset1, tConj tset2)"   | t1 :: "('a, 'b, 'c, 'd) Tree" and t2 :: "('a, 'b, 'c, 'd) Tree" where "x = (tNot t1__, tNot t2__)"   | f1 :: "'d" and \<phi>1 :: "'b" and f2 :: "'d" and \<phi>2 :: "'b" where "x = (tPred f1__ \<phi>1, tPred f2__ \<phi>2)"   | f1 :: "'d" and \<alpha>1 :: "'c" and t1 :: "('a, 'b, 'c, 'd) Tree" and f2 :: "'d" and \<alpha>2 :: "'c" and t2 :: "('a, 'b, 'c, 'd) Tree" where "x = (tAct f1 \<alpha>1 t1, tAct f2 \<alpha>2 t2)"   | v :: "('a, 'b, 'c, 'd) Tree" and va :: "('a, 'b, 'c, 'd) Tree set['a]" where "x = (tNot v__, tConj va__)"   | v :: "('a, 'b, 'c, 'd) Tree" and va :: "'d" and vb :: "'b" where "x = (tNot vb__, tPred vaa__ vba__)"   | v :: "('a, 'b, 'c, 'd) Tree" and va :: "'d" and vb :: "'c" and vc :: "('a, 'b, 'c, 'd) Tree" where "x = (tNot vc__, tAct vab__ vbb__ vca__)"   | v :: "'d" and va :: "'b" and vb :: "('a, 'b, 'c, 'd) Tree set['a]" where "x = (tPred vd__ vac__, tConj vbc__)"   | v :: "'d" and va :: "'b" and vb :: "('a, 'b, 'c, 'd) Tree" where "x = (tPred ve__ vad__, tNot vbd__)"   | v :: "'d" and va :: "'b" and vb :: "'d" and vc :: "'c" and vd :: "('a, 'b, 'c, 'd) Tree" where "x = (tPred vf__ vae__, tAct vbe__ vcb__ vda__)"   | v :: "'d" and va :: "'c" and vb :: "('a, 'b, 'c, 'd) Tree" and vc :: "('a, 'b, 'c, 'd) Tree set['a]" where "x = (tAct vg__ vaf__ vbf__, tConj vcc__)"   | v :: "'d" and va :: "'c" and vb :: "('a, 'b, 'c, 'd) Tree" and vc :: "('a, 'b, 'c, 'd) Tree" where "x = (tAct vh__ vag__ vbg__, tNot vcd__)"   | v :: "'d" and va :: "'c" and vb :: "('a, 'b, 'c, 'd) Tree" and vc :: "'d" and vd :: "'b" where "x = (tAct vi__ vah__ vbh__, tPred vce__ vdb__)"   | va :: "('a, 'b, 'c, 'd) Tree set['a]" and v :: "('a, 'b, 'c, 'd) Tree" where "x = (tConj vai__, tNot vj__)"   | va :: "'d" and vb :: "'b" and v :: "('a, 'b, 'c, 'd) Tree" where "x = (tPred vaj__ vbi__, tNot vk__)"   | va :: "'d" and vb :: "'c" and vc :: "('a, 'b, 'c, 'd) Tree" and v :: "('a, 'b, 'c, 'd) Tree" where "x = (tAct vak__ vbj__ vcf__, tNot vl__)"   | vb :: "('a, 'b, 'c, 'd) Tree set['a]" and v :: "'d" and va :: "'b" where "x = (tConj vbk__, tPred vm__ val__)"   | vb :: "('a, 'b, 'c, 'd) Tree" and v :: "'d" and va :: "'b" where "x = (tNot vbl__, tPred vn__ vam__)"   | vb :: "'d" and vc :: "'c" and vd :: "('a, 'b, 'c, 'd) Tree" and v :: "'d" and va :: "'b" where "x = (tAct vbm__ vcg__ vdc__, tPred vo__ van__)"   | vc :: "('a, 'b, 'c, 'd) Tree set['a]" and v :: "'d" and va :: "'c" and vb :: "('a, 'b, 'c, 'd) Tree" where "x = (tConj vch__, tAct vp__ vao__ vbn__)"   | vc :: "('a, 'b, 'c, 'd) Tree" and v :: "'d" and va :: "'c" and vb :: "('a, 'b, 'c, 'd) Tree" where "x = (tNot vci__, tAct vq__ vap__ vbo__)"   | vc :: "'d" and vd :: "'b" and v :: "'d" and va :: "'c" and vb :: "('a, 'b, 'c, 'd) Tree" where "x = (tPred vc vd, tAct v va vb)"
transp_onI: fixes A :: "'d set"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>x \<in> A; y \<in> A; z \<in> A; P x y; P y z\<rbrakk> \<Longrightarrow> P x z" shows "transp_on P A"
RInitInitThrowReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh'' :: "sheap"   and D :: "char list"   and Cs :: "char list list"   and e\<^sub>0 :: "char list exp"   and e\<^sub>1 :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>Throw a,(h', l', sh'),b'\<rangle>"   and "sh' C = \<lfloor>(sfs, i)\<rfloor>"   and "sh'' = sh'(C \<mapsto> (sfs, Error))"   and "P \<turnstile> \<langle>RI (D,Throw a) ; Cs \<leftarrow> e\<^sub>0,(h', l', sh''),b'\<rangle> \<rightarrow>* \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>RI (C,e) ; D # Cs \<leftarrow> e\<^sub>0,s,b\<rangle> \<rightarrow>* \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"
VxVxV_comp_eq_VVV_comp: shows "VxVxV.comp = VVV.comp"
distrib: fixes g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "finite {a. g a \<noteq> (0::'b)}"   and "finite {a. h a \<noteq> (0::'b)}" shows "(\<Sum>a. g a + h a) = Sum_any g + Sum_any h"
l_new_element_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_new_element_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs"
bind_def: fixes f :: "'b Heap"   and g :: "'b \<Rightarrow> 'a Heap" shows "f \<bind> g = Heap.Heap (\<lambda>h. case execute f h of None \<Rightarrow> None | Some (x, h') \<Rightarrow> execute (g x) h')"
dom_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.dom (src\<^sub>B \<mu>) = src\<^sub>B \<mu>"
V_simps_3: fixes \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and trn :: "('a, 'b, 'c) trans"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and tr :: "('a, 'b, 'c) trans list" assumes "\<phi> trn" shows "BD_Security.V \<phi> f (trn # tr) = f trn # BD_Security.V \<phi> f tr"
Nil_in_lists: fixes A :: "'b set" shows "[] \<in> lists A"
liminf_minus_ennreal: fixes v :: "nat \<Rightarrow> ennreal"   and u :: "nat \<Rightarrow> ennreal" assumes "\<And>n. v n \<le> u n" shows "liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
rh: fixes U :: "'a set set"   and V :: "'a set set" assumes "is_zariski_open U"   and "is_zariski_open V"   and "V \<subseteq> U" shows "ring_homomorphism (im_sheaf_morphisms U V) (local.im_sheaf U) (add_sheaf_spec U) (mult_sheaf_spec U) (zero_sheaf_spec U) (one_sheaf_spec U) (local.im_sheaf V) (add_sheaf_spec V) (mult_sheaf_spec V) (zero_sheaf_spec V) (one_sheaf_spec V)"
aGroup_def: fixes A :: "('c, 'd) aGroup_scheme" shows "aGroup A \<equiv> ((\<plusminus>\<^bsub>A\<^esub>) \<in> carrier A \<rightarrow> carrier A \<rightarrow> carrier A \<and> (\<forall>a b c. a \<in> carrier A \<longrightarrow> b \<in> carrier A \<longrightarrow> c \<in> carrier A \<longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c)) \<and> (\<forall>a b. a \<in> carrier A \<longrightarrow> b \<in> carrier A \<longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b = b \<plusminus>\<^bsub>A\<^esub> a)) \<and> (mop A \<in> carrier A \<rightarrow> carrier A \<and> (\<forall>a. a \<in> carrier A \<longrightarrow> -\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> a = \<zero>\<^bsub>A\<^esub>)) \<and> \<zero>\<^bsub>A\<^esub> \<in> carrier A \<and> (\<forall>a. a \<in> carrier A \<longrightarrow> \<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a)"
intros_32: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es :: "expr list"   and vs :: "val list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Copt :: "char list option"   and M :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<^sub>2\<rangle>"
wf_fdecl_def2: fixes G :: "prog"   and P :: "pname"   and fd :: "fdecl" shows "wf_fdecl G P fd = is_acc_type G P (type (snd fd))"
cmp_simps_5: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "C.cod (\<Phi>\<^sub>H (f, g)) = H (f \<star>\<^sub>B g)"
measure_bind: fixes f :: "'a \<Rightarrow> 'b measure"   and M :: "'a pmf"   and N :: "'b measure"   and X :: "'b set" assumes "measure_pmf.random_variable M (subprob_algebra N) f"   and "X \<in> sets N" shows "Sigma_Algebra.measure (measure_pmf M \<bind> f) X = measure_pmf.expectation M (\<lambda>x. Sigma_Algebra.measure (f x) X)"
hom: fixes x :: "'a"   and y :: "'a" shows "max (f x) (f y) = f (max x y)"
distrib: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "(a + b) * c = a * c + b * c"
triangle_right: shows "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
4: shows "to_fun (shift 2 (T\<^bsub>b\<^esub> g)) (a \<ominus> b) \<in> S"
single_valued_as_brE: fixes R :: "('a \<times> 'c) set" assumes "single_valued R" obtains \<alpha> :: "'a \<Rightarrow> 'c" and invar :: "'a \<Rightarrow> bool" where "R = br \<alpha> invar"
add_cancel_right_right: fixes a :: "'a"   and b :: "'a" shows "(a = a + b) = (b = (0::'a))"
limitV: fixes t :: "'a"   and n :: "nat" assumes "t \<in> S \<inter> V" shows "1 - (real k * \<delta> / 2) ^ n \<le> q n t"
lt_plus_lessE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p \<prec>\<^sub>t lt (p + q)" shows "lt p \<prec>\<^sub>t lt q"
False: shows "\<not> (p = 0 \<or> q = 0)"
hom_sub: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "eval R R id x (x \<ominus>\<^bsub>P\<^esub> y) = eval R R id x x \<ominus> eval R R id x y"
simps_2: fixes x :: "'a"   and xs :: "'a ppath"   and ys :: "'a ppath" shows "pp_fusion (ppath.Cons x xs) ys = ppath.Cons x (pp_fusion xs ys)"
lappend_LNil2: fixes xs :: "'a llist" shows "xs @\<^sub>l []\<^sub>l = xs"