hcju/isabelleps huggingface dataset:
try_spmf_bind_spmf_lossless: fixes p :: "'h spmf"   and f :: "'h \<Rightarrow> 'i spmf"   and q :: "'i spmf" assumes "lossless_spmf p" shows "TRY p \<bind> f ELSE q = p \<bind> (\<lambda>x. TRY f x ELSE q)"
star_inductl_star: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y\<^sup>\<star> \<le> y\<^sup>\<star>" shows "x\<^sup>\<star> \<le> y\<^sup>\<star>"
reduction_signs_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list"   and matr :: "rat mat" shows "reduction_signs_R p qs signs subsets matr = take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets matr))"
length_helper2: fixes p :: "nat"   and q :: "nat"   and m :: "bit list"   and M :: "bit list" assumes "prime p"   and "prime q"   and "MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))) \<noteq> []"   and "length (sha1 M) + sLen + 16 \<le> length (nat_to_bv (p * q)) - Suc 0" shows "length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))))) = roundup (length (nat_to_bv (p * q)) - Suc 0) 8 * 8 - 168"
subsetD: fixes A :: "'q set"   and B :: "'q set"   and c :: "'q" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
assms: shows "S closed_in (\<Gamma> @ VarB X Q # \<Delta>)"
is_normal: fixes n :: "nat"   and v :: "complex mat" assumes "state n v" shows "\<parallel>col v 0\<parallel> = 1"
inverse_unique: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<otimes> b = \<one>" shows "inv a = b"
edom_env_delete: fixes x :: "'b"   and m :: "'b \<Rightarrow> 'c" shows "edom (env_delete x m) = edom m - {x}"
of_real_Re: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "nat" assumes "real_dcharacter n \<chi>" shows "complex_of_real (Re (\<chi> k)) = \<chi> k"
emeasure_PiM_emb: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and J :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"   and "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure (Pi\<^sub>M I M) (prod_emb I M J (Pi\<^sub>E J A)) = (\<Prod>i\<in>J. emeasure (M i) (A i))"
LLL_inv_m_imp_reduced: fixes fs :: "int vec list" assumes "LLL_invariant True m fs" shows "gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> m"
Rep_process_inject: fixes x :: "'a process"   and y :: "'a process" shows "(Rep_process x = Rep_process y) = (x = y)"
HFun_Sigma_SeqAppendP: fixes f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm" shows "{SeqAppendP f1 k1 f2 k2 g, HFun_Sigma f1, HFun_Sigma f2, OrdP k1} \<turnstile> HFun_Sigma g"
rel_simps_71: shows "\<not> (1::'a) < (1::'a)"
all_sat: shows "\<forall>qi\<in>Q. calculus.saturated Inf (Red_I_q qi) N"
zero_divisorI: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.mult_zero times zero"   and "b \<noteq> zero"   and "times a b = zero" shows "mult_zero.zero_divisor times zero a"
sum_list_eq_nthI: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs"   and "\<And>j. \<lbrakk>j < length xs; j \<noteq> i\<rbrakk> \<Longrightarrow> xs ! j = (0::'b)" shows "sum_list xs = xs ! i"
mod256: fixes d :: "nat"   and c :: "nat"   and b :: "nat"   and a :: "nat" shows "(d + 256 * c + 65536 * b + 16777216 * a) mod 256 = d mod 256"
smult_pCons: fixes a :: "'b"   and b :: "'b"   and p :: "'b poly" shows "Polynomial.smult a (pCons b p) = pCons (a * b) (Polynomial.smult a p)"
approx_add_mono2: fixes b :: "'a star"   and c :: "'a star"   and a :: "'a star" assumes "b \<approx> c" shows "b + a \<approx> c + a"
neg_add_closed: fixes g :: "'g"   and h :: "'g" assumes "g \<in> G"   and "h \<in> G" shows "- g + h \<in> G"
exists_finite_inconsistent: fixes C :: "'b fm set set"   and V :: "'b fm set"   and p :: "'b fm" assumes "finite_char C"   and "V \<union> {\<^bold>\<not> p} \<notin> C" shows "\<exists>W. W \<union> {\<^bold>\<not> p} \<subseteq> V \<union> {\<^bold>\<not> p} \<and> (\<^bold>\<not> p) \<notin> W \<and> finite W \<and> W \<union> {\<^bold>\<not> p} \<notin> C"
same_set_semiJoin: fixes x :: "nat set \<times> 'b option list set"   and other :: "nat set \<times> 'b option list" shows "fst (New_max.semiJoin x other) = fst x"
rtranclp_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "'a \<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"
scene_override_union: fixes X :: "'a scene"   and Y :: "'a scene"   and S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a" assumes "X ##\<^sub>S Y" shows "S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on (X \<squnion>\<^sub>S Y) = S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on X \<oplus>\<^sub>S S\<^sub>2 on Y"
iMODb_trans: fixes y :: "nat"   and x :: "nat"   and m :: "nat"   and c :: "nat"   and z :: "nat"   and c' :: "nat" assumes "y \<in> [ x, mod m, c ]"   and "z \<in> [ y, mod m, c' ]"   and "m * c' \<le> x + m * c - y" shows "z \<in> [ x, mod m, c ]"
hyps_10: fixes b :: "trm" assumes "e\<^sub>1'[x::=e\<^sub>2'] \<Down> b" shows "e' = b"
adj_map: fixes C :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "domain.chamber D"   and "C \<sim> D" shows "f ` C \<sim> f ` D"
biessentially_surjective_on_objects: fixes a' :: "'c" assumes "D.obj a'" shows "\<exists>a. C.obj a \<and> D.equivalent_objects (G.map\<^sub>0 a) a'"
bounded_two_points: fixes S :: "'c set" shows "bounded S = (\<exists>e. \<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> e)"
assms_4: shows "f \<in> UNIV \<rightarrow> {- 1..1}"
SN_on_Image_rtrancl_conv: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "SN_on r A = SN_on r (r\<^sup>* `` A)"
simps_3: fixes S1 :: "type_scheme"   and S2 :: "type_scheme" shows "free_tv (S1 =-> S2) = free_tv S1 \<union> free_tv S2"
wordinterval_setminus'_rr_set_eq: fixes s :: "'b word"   and e :: "'b word"   and ms :: "'b word"   and me :: "'b word" shows "wordinterval_to_set (wordinterval_setminus' (WordInterval s e) (WordInterval ms me)) = wordinterval_to_set (WordInterval s e) - wordinterval_to_set (WordInterval ms me)"
additive_sum: fixes M :: "'b measure"   and f :: "'b set \<Rightarrow> ennreal"   and S :: "'c set"   and A :: "'c \<Rightarrow> 'b set" assumes "positive (sets M) f"   and "Measure_Space.additive (sets M) f"   and "finite S"   and "A ` S \<subseteq> sets M"   and "disjoint_family_on A S" shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union> (A ` S))"
that: fixes z :: "'a" assumes "z \<in> open_segment a b"   and "(\<forall>w\<in>closed_segment a b. f w \<le> f z) \<or> (\<forall>w\<in>closed_segment a b. f z \<le> f w)" shows "thesis"
insert_Diff_if: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
eql_cong: fixes H :: "fmla set"   and t :: "trm"   and t' :: "trm"   and u :: "trm"   and u' :: "trm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> t EQ u IFF t' EQ u'"
the_g: shows "(THE fa. \<forall>a. f v $ a = (\<Sum>x\<in>UNIV. fa x * Y $ x $ a)) = g"
cbv_in_ctx: fixes t :: "lam"   and t' :: "lam"   and E :: "ctx" assumes "t \<longrightarrow>cbv t'" shows "E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>"
assms_1: shows "u\<^sub>1 \<preceq>\<^sub>F\<^sub>I w"
is_isoI: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "h \<in> Group.iso G H" shows "G \<cong> H"
upper_asymptotic_density_incseq_Union: fixes A :: "nat \<Rightarrow> nat set"   and l :: "real" assumes "\<And>n. upper_asymptotic_density (A n) \<le> l"   and "incseq A" shows "\<exists>B. upper_asymptotic_density B \<le> l \<and> (\<forall>n. \<exists>N. A n \<inter> {N..} \<subseteq> B)"
gc_initial_state_def: fixes s :: "('a, 'b, 'c, 'd) Model.local_state" shows "gc_initial_state s = (fM s = initial_mark \<and> phase s = ph_Idle \<and> ghost_honorary_grey s = {} \<and> W s = {})"
assms: fixes X :: "'a set" assumes "X \<subseteq> A"   and "X \<noteq> {}" shows "\<exists>x\<in>X. \<forall>y\<in>X. \<not> y \<sqsubset> x"
has_method_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and M :: "char list" shows "P \<turnstile> C has M \<equiv> \<exists>Ts T m D. P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"
red_hext_incr: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and 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" assumes "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "h \<unlhd> h'"
nxt_wait: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev \<phi> xs" shows "(nxt ^^ wait \<phi> xs) \<phi> xs"
ltlc_to_ltln'_correct_4: fixes \<phi> :: "'b ltlc" shows "size (ltlc_to_ltln' False \<phi>) \<le> 2 * size \<phi>"
cond_A_def: fixes xs :: "nat list"   and ys :: "nat list" shows "cond_A xs ys = (\<forall>x\<in>set xs. x \<le> maxne0 ys b)"
assms_2: shows "Q \<sqsubseteq> Q ;; P"
cases_scheme: fixes r :: "'a state_t_scheme" obtains sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool" and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool" and current :: "thread_id_t" and obj :: "obj_id_t \<Rightarrow> obj_t" and thread :: "thread_id_t \<Rightarrow> thread_t" and more :: "'a" where "r = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr>"
init_u: fixes t :: "real" assumes "t \<le> 0" shows "u t = s\<^sub>e"
value_gt_false_Some: fixes a :: "value option"   and b :: "value option" assumes "value_gt a b = false" shows "(\<exists>x. a = Some x) \<and> (\<exists>y. b = Some y)"
distinct_14: fixes x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly"   and x1 :: "'b" shows "CN x81 x82 x83 \<noteq> (x1)\<^sub>p"
Block1Some: fixes V :: "nat"   and x :: "'addr val list"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and T :: "ty"   and v :: "'addr val"   and e :: "'addr expr1"   and h :: "'heap" assumes "V < length x" shows "uf,P,t \<turnstile>1 \<langle>{V:T=\<lfloor>v\<rfloor>; e},(h, x)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>{V:T=None; e},(h, x[V := v])\<rangle>"
assms_2: shows "\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
scale_left_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) vec" shows "(a + b) *s x = a *s x + b *s x"
induct: fixes P :: "(node \<times> node) list \<Rightarrow> node list \<Rightarrow> node list \<Rightarrow> bool"   and a0 :: "(node \<times> node) list"   and a1 :: "node list"   and a2 :: "node list" assumes "\<And>g ys. P g [] ys"   and "\<And>g x xs ys. \<lbrakk>List.member ys x \<Longrightarrow> P g xs ys; \<not> List.member ys x \<Longrightarrow> P g (nexts g x) (x # ys); \<not> List.member ys x \<Longrightarrow> P g xs (dfs2 g (nexts g x) (x # ys))\<rbrakk> \<Longrightarrow> P g (x # xs) ys" shows "P a0 a1 a2"
locally_sound_modes_invariant: fixes c :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "locally_sound_mode_use \<langle>c, mds, mem\<rangle>"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "locally_sound_mode_use \<langle>c', mds', mem'\<rangle>"
zero_order_1: fixes x :: "'h multiset" shows "{#} \<subseteq># x"
lf: shows "list_all (\<lambda>i. i < length signs) (fst I)"
omega_complete_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" shows "omega_complete le lub \<equiv> porder le \<and> omega_complete_axioms le lub"
Close1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and y :: "name"   and Q' :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'"   and "Q \<longmapsto> a<\<nu>y> \<prec> Q'"   and "y \<sharp> P" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> <\<nu>y>(P'[x::=y] \<parallel> Q')"
less_add_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less a (plus a one)"
ord_of_nat_eq_initial: fixes n :: "nat" shows "ord_of_nat n = set (ord_of_nat ` {..<n})"
scale_zero_left: fixes x :: "'a \<Rightarrow> real" assumes "x \<in> diff_fun_space" shows "0 *\<^sub>R x = 0"
absorb_iff2: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry" shows "(b \<preceq> a) = (linordered_monoid.min a b = b)"
naturality: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "\<sigma> (CC.cod f) \<cdot> T f = ToS.map f \<cdot> \<sigma> (CC.dom f)"
I_subst: fixes f :: "atom fm"   and k :: "nat"   and xs :: "'a list" assumes "qfree f" shows "DLO.I (subst f k) xs = DLO.I f (xs ! k # xs)"
hmem_shift_add_iff: fixes delta :: "hf"   and n :: "hf"   and y :: "hf"   and f :: "hf" shows "(\<langle>delta @+ n, y\<rangle> \<^bold>\<in> shift f delta) = (\<langle>n, y\<rangle> \<^bold>\<in> f)"
PPs_UnI: fixes tx :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and ty :: "'b \<Rightarrow>\<^sub>0 nat"   and Y :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "tx \<in> .[X]"   and "ty \<in> .[Y]"   and "t = tx + ty" shows "t \<in> .[X \<union> Y]"
simps_2: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "f^(Suc n) = f \<circ> f^n"
assms_3: shows "A\<^sub>F \<sharp>* \<Phi>"
assms_6: shows "(\<Sum>x | X x \<noteq> (0::'a). X x *s x) = (\<Sum>x | Y x \<noteq> (0::'a). Y x *s x)"
XD_smallest_def: fixes ds :: "'b set"   and X :: "'a set" shows "TruePref_tax.XD_smallest ds X = - (TruePref_tax.XH_largest ds X - X)"
distinct_2: fixes name' :: "name"   and name :: "name" shows "BoundOutputS name' \<noteq> InputS name"
mcont_const: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'b" assumes "class.ccpo Sup less_eq less" shows "mcont luba orda Sup less_eq (\<lambda>x. c)"
nn_integral_add: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x + g x \<partial>M = integral\<^sup>N M f + integral\<^sup>N M g"
9: shows "prim_P w v e\<^sup>T * (w \<sqinter> - prim_EP w v e)\<^sup>+ * e = bot"
right_idem: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "bounded_quasi_semilattice f top bot normalize" shows "f (f a b) b = f a b"
nb: shows "\<forall>(v, a)\<in>dom (\<phi>\<^sub>S \<Psi> s). \<forall>(v, a')\<in>dom (\<phi>\<^sub>S \<Psi> s). (\<phi>\<^sub>S \<Psi> s) (v, a) = Some True \<and> (\<phi>\<^sub>S \<Psi> s) (v, a') = Some True \<longrightarrow> (v, a) = (v, a')"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree"   and a1 :: "('b \<times> color) tree" assumes "\<And>t. P \<langle>\<rangle> t"   and "\<And>v va vb. P \<langle>v, va, vb\<rangle> \<langle>\<rangle>"   and "\<And>t1 a t2 t3 c t4. P t2 t3 \<Longrightarrow> P (R t1 a t2) (R t3 c t4)"   and "\<And>t1 a t2 t3 c t4. P t2 t3 \<Longrightarrow> P (B t1 a t2) (B t3 c t4)"   and "\<And>v vc vb t2 a t3. P (B v vc vb) t2 \<Longrightarrow> P (B v vc vb) (R t2 a t3)"   and "\<And>t1 a t2 v vc vb. P t2 (B v vc vb) \<Longrightarrow> P (R t1 a t2) (B v vc vb)" shows "P a0 a1"
comp_red_basis_aux_lt: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_minimal_basis (set (xs @ ys))"   and "distinct (xs @ ys)" shows "lt ` set (xs @ ys) = lt ` set (comp_red_basis_aux xs ys)"
card_f: shows "card V = card (f ` d_reachable M (initial M))"
converse_rtrancl_on_induct: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> rtrancl_on F r"   and "b \<in> F \<Longrightarrow> P b"   and "\<And>x y. \<lbrakk>(x, y) \<in> r; (y, b) \<in> rtrancl_on F r; x \<in> F; y \<in> F; P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
FAccRedsNull: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<rangle> \<rightarrow>* \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s'\<rangle>"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "\<And>x. hom x = (1::'b) \<Longrightarrow> x = (1::'a)" shows "one_hom_1_axioms hom"
snd_conv: fixes x1 :: "'c"   and x2 :: "'a" shows "snd (x1, x2) = x2"
epm_2: fixes w :: "'a"   and v :: "'a"   and e :: "'a" assumes "regular (prim_EP w v e)"   and "vector v" shows "w \<sqinter> - prim_EP w v e \<squnion> prim_P w v e \<squnion> prim_E w v e = w"
prems_7: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
seqI': fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "Cop.in_hom f a b"   and "Cop.in_hom g b c" shows "Cop.seq g f"
abs_summable_on_reindex_bij_betw: fixes g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "bij_betw g A B" shows "((\<lambda>x. f (g x)) abs_summable_on A) = (f abs_summable_on B)"
getNeg_append: fixes as :: "'b negation_type list"   and bs :: "'b negation_type list" shows "getNeg (as @ bs) = getNeg as @ getNeg bs"
top_Err: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Semilat.top (le r) Err"
Nle_def: shows "(\<le>\<^sub>N) = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
pair_spmf_alt_def: fixes p :: "'c spmf"   and q :: "'d spmf" shows "pair_spmf p q = p \<bind> (\<lambda>x. q \<bind> (\<lambda>y. return_spmf (x, y)))"
invrules_3: fixes T' :: "nat \<Rightarrow> nat"   and A' :: "nat \<Rightarrow> nat set" assumes "lb T' A' j" shows "real (makespan T) \<le> 3 / 2 * real (makespan T')"
Greater_unbounded: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "finite X"   and "Regions.valid_region X k I r"   and "c \<in> X"   and "I c = Regions.intv.Greater (k c)" shows "unbounded c (Regions.region X I r)"
gram_schmidt_hd: fixes w :: "'b vec"   and ws :: "'b vec list" assumes "w \<in> carrier_vec n" shows "hd (gram_schmidt n (w # ws)) = w"
enforce_\<I>_gpv_def: shows "enforce_\<I>_gpv \<equiv> \<lambda>uu uua. corec_gpv (\<lambda>(\<I>, gpv). map_spmf (map_generat id id ((\<circ>) (\<lambda>uu. Inr (\<I>, uu)))) (map_spmf (\<lambda>generat. case generat of Pure x \<Rightarrow> Pure x | IO out rpv \<Rightarrow> IO out (\<lambda>input. if input \<in> responses_\<I> \<I> out then rpv input else Fail)) (enforce_spmf (pred_generat \<top> (\<lambda>x. x \<in> outs_\<I> \<I>) \<top>) (the_gpv gpv)))) (uu, uua)"
subspace_sum: fixes A :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and B :: "'b set" assumes "A \<subseteq> S"   and "\<forall>x. f x \<in> S"   and "subspace A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"
assms_2: fixes n :: "nat" assumes "0 < n" shows "u n \<le> real n * a"
image_Fpow_mono: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A \<subseteq> B" shows "(`) f ` Fpow A \<subseteq> Fpow B"
pf_nonneg_mat_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "pf_nonneg_mat A \<equiv> non_neg_mat A"
lfp_bound: shows "\<alpha> bot \<le> lfp g"
sel_3: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "cl (Node x21 x22 x23 x24 x25) = x22"
N_of_norm_def: shows "N_of_norm = quasinorm_of (case (1, norm) of (x, y) \<Rightarrow> (x, \<lambda>x. ennreal (y x)))"
inject: fixes a :: "'k"   and b :: "'l"   and a' :: "'k"   and b' :: "'l" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
perm_app: fixes pi :: "(name \<times> name) list"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "pi \<bullet> f x = (pi \<bullet> f) (pi \<bullet> x)"
app: fixes t\<^sub>1 :: "'a"   and t\<^sub>2 :: "'a" assumes "t = app t\<^sub>1 t\<^sub>2"   and "no_abs t\<^sub>1"   and "no_abs t\<^sub>2" shows "thesis"
Pi_pmf_cong: fixes A :: "'b set"   and A' :: "'b set"   and dflt :: "'c"   and dflt' :: "'c"   and f :: "'b \<Rightarrow> 'c pmf"   and f' :: "'b \<Rightarrow> 'c pmf" assumes "A = A'"   and "dflt = dflt'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "Pi_pmf A dflt f = Pi_pmf A' dflt' f'"
bounded_B: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "integral\<^sup>N (count_space (\<^bold>E `` {x})) (weight \<Gamma>) < \<top>"
finite_measure_mono: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b pmf" assumes "A \<subseteq> B"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M A \<le> measure_pmf.prob M B"
select_convs_3: fixes routing_match :: "'b prefix_match"   and metric :: "nat"   and routing_action :: "'b routing_action"   and more :: "'c" shows "routing_action \<lparr>routing_match = routing_match, metric = metric, routing_action = routing_action, \<dots> = more\<rparr> = routing_action"
dom_img_disj: shows "subst_domain \<sigma>1 \<inter> range_vars \<theta>2 = {}"
non_interference_step: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "get_S (cpu_reg_val PSR s1) = 0 \<and> good_context s1 \<and> get_delayed_pool s1 = [] \<and> get_trap_set s1 = {} \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> get_delayed_pool s2 = [] \<and> get_trap_set s2 = {} \<and> good_context s2 \<and> low_equal s1 s2" shows "\<exists>t1 t2. Some t1 = NEXT s1 \<and> Some t2 = NEXT s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2"
zero_infty_def: shows "0 \<equiv> Num (0::'a)"
new_AddrD: fixes h :: "loc \<Rightarrow> obj option"   and a :: "loc" assumes "new_Addr h = Some a" shows "h a = None"