hcju/isabelleps huggingface dataset:
D_def: shows "D = (\<lambda>x. (- fst x, - snd x)) -` A"
simps: fixes a1 :: "collection_ternop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'b OCL_Types.type"   and a4 :: "'a OCL_Types.type"   and a5 :: "'a OCL_Types.type" shows "collection_ternop_type a1 a2 a3 a4 a5 = ((\<exists>\<sigma> \<rho> \<tau>. a1 = InsertAtOp \<and> a2 = OrderedSet \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> a5 = OrderedSet \<tau> \<and> \<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1] \<and> \<rho> \<le> \<tau>) \<or> (\<exists>\<sigma> \<rho> \<tau>. a1 = InsertAtOp \<and> a2 = Sequence \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> a5 = Sequence \<tau> \<and> \<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1] \<and> \<rho> \<le> \<tau>) \<or> (\<exists>\<sigma> \<rho> \<tau>. a1 = SubOrderedSetOp \<and> a2 = OrderedSet \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> a5 = OrderedSet \<tau> \<and> \<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1] \<and> \<rho> = UnlimitedNatural[1]\<midarrow>Integer[1]) \<or> (\<exists>\<sigma> \<rho> \<tau>. a1 = SubSequenceOp \<and> a2 = Sequence \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> a5 = Sequence \<tau> \<and> \<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1] \<and> \<rho> = UnlimitedNatural[1]\<midarrow>Integer[1]))"
valid_prefixes_split: fixes r :: "('b, 'c) routing_rule_scheme"   and rs :: "('b, 'c) routing_rule_scheme list" assumes "valid_prefixes (r # rs)" shows "valid_prefix (routing_match r) \<and> valid_prefixes rs"
singleton: fixes x :: "'k" shows "\<Sum>\<^sub># {#x#} = x"
confluentp_imp_semiconfluentp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "confluentp r" shows "semiconfluentp r"
weight_w0: fixes t :: "('f, 'c) Term.term" shows "w0 \<le> weight t"
sup_ident: fixes x :: "'a" shows "x \<squnion> \<bottom> = x"
ih_4: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
C\<^sub>2: shows "\<^bold>\<forall>x xa xb. E ((x \<cdot> xa) \<cdot> xb) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (x \<cdot> (xa \<cdot> xb))) \<^bold>\<or> (x \<cdot> xa) \<cdot> xb \<noteq> x \<cdot> (xa \<cdot> xb)) \<^bold>\<leftarrow> E (x \<cdot> xa) \<^bold>\<and> E (xa \<cdot> xb)"
coboundedI1: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<le> a" shows "c \<le> max a b"
simps_1: fixes uu :: "'b \<Rightarrow> bool" shows "find_index uu [] = 0"
degree_0_id: fixes p :: "'a poly" assumes "degree p = 0" shows "[:coeff p 0:] = p"
1: shows "Maps.commutative_square SPN_fgh.\<nu>.leg0 SPN_fgh.\<pi>.leg1 SPN_fgh.Prj\<^sub>0\<^sub>1 SPN_fgh.Prj\<^sub>0"
orthogonalD_in: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "orthogonal \<Delta> f S"   and "edge \<Delta> x y"   and "x \<notin> S"   and "y \<in> S" shows "f (x, y) = 0"
qGoodInp_def: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" shows "qGoodInp qinp \<equiv> liftAll qGood qinp \<and> |{i. qinp i \<noteq> None}| <o |UNIV|"
restrict_index_in: fixes i :: "nat"   and X :: "'c option list"   and I :: "nat set" assumes "i < length X"   and "i \<in> I" shows "restrict I X ! i = X ! i"
weakly_preserves_objects: fixes a :: "'b" assumes "B.obj a" shows "C.isomorphic (\<tau>.G.map\<^sub>0 a) (G a)"
zero_cvec_z1_zero_iff: fixes z1 :: "complex"   and z2 :: "complex" assumes "(z1, z2) \<noteq> vec_zero" shows "((z1, z2) \<approx>\<^sub>v 0\<^sub>v) = (z1 = 0)"
f_shrink_1: fixes xs :: "'d message_af list" shows "xs \<div> Suc 0 = xs"
infinite_chain_lnth_rel: fixes c :: "'b llist"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "nat" assumes "\<not> lfinite c"   and "chain r c" shows "r (lnth c i) (lnth c (Suc i))"
4: assumes "monomial_decomp ps" shows "monomial_decomp qs"
r_def: fixes tree :: "'b tree" shows "r tree = (case tree of Leaf \<Rightarrow> 0 | Node x21 x22 x23 x24 x25 \<Rightarrow> x22)"
basis_card_eq_dim: fixes B :: "('b, 'c) vec set"   and V :: "('b, 'c) vec set" assumes "B \<subseteq> V"   and "V \<subseteq> vec.span B"   and "vec.independent B" shows "card B = vec.dim V"
leh: fixes x :: "'n"   and k :: "nat" assumes "x \<in> S" shows "norm (min (max (- h x) (f k x)) (h x)) \<le> h x"
fun_pow_iterates_above: fixes k :: "nat"   and a :: "'a" shows "(f ^^ k) a \<in> iterates_above a"
Inf_to_Sup_var: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and dual :: "'b \<Rightarrow> 'b"   and X :: "'b set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Inf X = dual (Sup (dual ` X))"
ereal_liminf_add_mono: fixes u :: "nat \<Rightarrow> ereal"   and v :: "nat \<Rightarrow> ereal" assumes "\<not> (liminf u = \<infinity> \<and> liminf v = - \<infinity> \<or> liminf u = - \<infinity> \<and> liminf v = \<infinity>)" shows "liminf u + liminf v \<le> liminf (\<lambda>n. u n + v n)"
assms: shows "LIM x F. norm (f x) :> at_top"
restrict_extend_indets: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "restrict_indets (extend_indets p) = p"
assms_4: shows "b \<noteq> \<zero>\<^bsub>K [X]\<^esub>"
Ord_plus_Ord_closed_augend: fixes \<alpha> :: "V"   and \<gamma> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<gamma>"   and "\<alpha> + \<beta> \<in>\<^sub>\<circ> \<gamma>" shows "\<alpha> \<in>\<^sub>\<circ> \<gamma>"
snoc: shows "prod_list (map f xs) = prod_list (map f (nths xs A)) * prod_list (map f (nths xs (- A)))"
ext_inject: fixes ob :: "'b set"   and ar :: "'c set"   and dom :: "'c \<Rightarrow> 'b"   and cod :: "'c \<Rightarrow> 'b"   and id :: "'b \<Rightarrow> 'c"   and comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d"   and ob' :: "'b set"   and ar' :: "'c set"   and dom' :: "'c \<Rightarrow> 'b"   and cod' :: "'c \<Rightarrow> 'b"   and id' :: "'b \<Rightarrow> 'c"   and comp' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more' :: "'d" shows "(\<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr> = \<lparr>ob = ob', ar = ar', dom = dom', cod = cod', id = id', comp = comp', \<dots> = more'\<rparr>) = (ob = ob' \<and> ar = ar' \<and> dom = dom' \<and> cod = cod' \<and> id = id' \<and> comp = comp' \<and> more = more')"
assms_2: shows "\<forall>eps>0. \<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. extended_distance (u n) (u m) < eps"
timpls_transformable_to_trans: fixes TI :: "('e set \<times> 'e set) list"   and s :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and u :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "\<forall>(a, b)\<in>(set TI)\<^sup>+. a \<noteq> b \<longrightarrow> (a, b) \<in> set TI"   and "timpls_transformable_to TI s t"   and "timpls_transformable_to TI t u" shows "timpls_transformable_to TI s u"
compactin_absolute: fixes X :: "'b topology"   and S :: "'b set" shows "compactin (subtopology X S) S = compactin X S"
indets_monomial: fixes c :: "'b"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'b)" shows "indets (monomial c t) = keys t"
le_D_adm: fixes u :: "'a \<Rightarrow> 'b process"   and v :: "'a \<Rightarrow> 'b process" assumes "cont u"   and "monofun v" shows "adm (\<lambda>x. u x \<sqsubseteq>\<^sub>D v x)"
s_Id: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and uu :: "mode" shows "\<lless>\<pi>, e\<ggreater> uu\<rightarrow>0 \<lless>\<pi>, e\<ggreater>"
ccExp_e: fixes a :: "Arity" shows "ccExp_syn a e = \<bottom>"
of_hypnat_le_iff: fixes m :: "nat star"   and n :: "nat star" shows "(of_hypnat m \<le> of_hypnat n) = (m \<le> n)"
init_fin_\<tau>divergeD: fixes t :: "'t"   and status :: "status"   and x :: "'x"   and m :: "'m" assumes "\<tau>trsys.\<tau>diverge (init_fin t) init_fin_\<tau>move ((status, x), m)" shows "\<tau>diverge t (x, m) \<and> status = Running"
hom: shows "homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. a)"
inductive_on_imp_accessible_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "inductive_on P A" shows "\<forall>x\<in>A. accessible_on P A x"
Aone: fixes s :: "'a"   and t :: "'a" assumes "M' s"   and "(s, t) \<in> A" shows "(s, t) \<in> (P' \<union> Pw' \<union> R' \<union> Rw') \<inter> {(s, t). M' t} \<and> M' t"
z: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A" shows "perron_frobenius.z A \<in> perron_frobenius.Y A"
dist_nns: fixes kdt :: "'b kdt"   and p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and n :: "nat" assumes "invar kdt"   and "sorted_wrt_dist p ps"   and "set ps \<inter> set_kdt kdt = {}"   and "distinct ps"   and "0 < n" shows "\<forall>q\<in>set_kdt kdt \<union> set ps - set (nearest_nbors n ps p kdt). dist (last (nearest_nbors n ps p kdt)) p \<le> dist q p"
discrete: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and a :: "'a"   and b :: "'a" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less a b = less_eq (plus a one) b"
T_on_embed: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and s0 :: "'a"   and qs :: "'b list"   and A :: "('a \<Rightarrow> 'd) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'd)" shows "real (On_Off.T step t s0 qs (On_Off.off2 step A (s0, fst A s0) qs)) = On_Off.T_on_rand' step t (On_Off.embed A) (fst (On_Off.embed A) s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs"
powr_one: fixes x :: "real" assumes "0 \<le> x" shows "x powr 1 = x"
char_of_comp_of_char: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "unique_euclidean_semiring_with_bit_shifts.char_of plus times one bit \<circ> comm_semiring_1.of_char times one plus zero = id"
in_these_eq: fixes x :: "'a"   and A :: "'a option set" shows "(x \<in> Option.these A) = (\<lfloor>x\<rfloor> \<in> A)"
remove1_append: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "remove1 x (xs @ ys) = (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
asymp_greater: shows "asymp (\<lambda>a b. cmp a b = Greater)"
construct_NofI_R_def: fixes p :: "real poly"   and I1 :: "real poly list"   and I2 :: "real poly list" shows "construct_NofI_R p I1 I2 = (let new_p = sum_list (map power2 (p # I1)) in rat_of_int (changes_R_smods new_p (pderiv new_p * prod_list I2)))"
S_def: shows "S \<equiv> {(d1, d2). d1 dvd gcd a m \<and> d2 dvd gcd k b}"
intro: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "symmetric A equiv"   and "reflexive A equiv" shows "tolerance A equiv"
runit_char_2: fixes f :: "'b" assumes "D.ide f" shows "D.R \<r>\<^sub>D[f] = (f \<star>\<^sub>D \<i>\<^sub>D[src\<^sub>D f]) \<cdot>\<^sub>D \<a>\<^sub>D[f, src\<^sub>D f, src\<^sub>D f]"
prim_recfn_total: fixes f :: "recf"   and n :: "nat" assumes "prim_recfn n f" shows "Partial_Recursive.total f"
map_type_eq_def: fixes uu :: "'a itself"   and uv :: "'b itself" shows "uu \<rightarrow>\<^sub>n\<^sub>t uv \<equiv> True"
simps_2: fixes hs :: "'b tree"   and uu :: "'b" shows "del_min \<langle>hs, uu, \<langle>\<rangle>\<rangle> = pass\<^sub>2 (pass\<^sub>1 hs)"
Phase1or2Write_HInv4a1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Phase1or2Write s s' p d"   and "HInv4a1 s q" shows "HInv4a1 s' q"
dom_el_idem: fixes x :: "'a" shows "ad (ad x) \<cdot> ad (ad x) = ad (ad x)"
eq_diff_eq: fixes V :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "(x = z - y) = (x + y = z)"
between_empty_imp_iprev_eq: fixes n :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "n \<in> A"   and "iprev n A < n"   and "n \<in> B"   and "iprev n A \<in> B"   and "B \<down>> iprev n A \<down>< n = {}" shows "iprev n B = iprev n A"
cancel_less: fixes x :: "ordinal"   and y :: "ordinal" shows "(F x < F y) = (x < y)"
x: shows "pvalid p x \<and> next_symbol x = Some (terminal_of_token u)"
all_fields_in_has_fields: fixes P :: "'a prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and D :: "String.literal"   and D' :: "String.literal"   and fs :: "fdecl list"   and ms :: "'a option mdecl list"   and F :: "String.literal"   and Tm :: "ty \<times> fmod" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C \<preceq>\<^sup>* D"   and "class P D = \<lfloor>(D', fs, ms)\<rfloor>"   and "(F, Tm) \<in> set fs" shows "((F, D), Tm) \<in> set FDTs"
abs_diff_triangle_ineq2: fixes a :: "'y"   and b :: "'y"   and c :: "'y" shows "\<bar>a - b\<bar> \<le> \<bar>a - c\<bar> + \<bar>c - b\<bar>"
add_8: fixes ii :: "nat" assumes "ii \<le> m" shows "d fs' ii = d fs ii"
induct: fixes P :: "'a list list \<Rightarrow> bool"   and a0 :: "'a list list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>v vb vc. P (merge_pairs key (v # vb # vc)) \<Longrightarrow> P (v # vb # vc)" shows "P a0"
aff_dim_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "aff_dim S \<le> aff_dim T"
br_step_in_termrel: fixes \<Sigma> :: "'d set \<times> 'd set"   and \<delta> :: "('d, 'e) ta_rule set"   and \<Sigma>' :: "'d set \<times> 'd set" assumes "\<Sigma> \<in> br_cond"   and "\<Sigma> \<in> br_invar \<delta>"   and "(\<Sigma>, \<Sigma>') \<in> br_step \<delta>" shows "(\<Sigma>', \<Sigma>) \<in> br_termrel \<delta>"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "slength\<cdot>(x :# xs) = slength\<cdot>xs + 1"
le_transD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "le_trans t u"   and "sound P" shows "t P \<tturnstile> u P"
less_numeral_extra_4: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one one"
fzero_def: shows "fzero = COERCE(udom\<cdot>'d, 'c\<cdot>'d)\<cdot>zeroU"
arr_char: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" shows "A2_B.arr f = (f \<noteq> A2_B.Null \<and> A2_B.Dom f \<in> Collect (functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B)) \<and> A2_B.Cod f \<in> Collect (functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B)) \<and> A2_B.Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (A2_B.Dom f) (A2_B.Cod f)))"
ensure_pre_insertion_validity_pure: fixes node :: "(_) node_ptr"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr option"   and h :: "(_) heap" shows "pure (a_ensure_pre_insertion_validity node ptr child) h"
in_q64_code: fixes x :: "nat" shows "(x mod 64 \<in> q64) = q64_array !! (x mod 64)"
4: shows "\<forall>x y. (case x of (e, fe) \<Rightarrow> \<lambda>(d, fd). fe + bot \<le> fd + bot) y \<or> (case y of (e, fe) \<Rightarrow> \<lambda>(d, fd). fe + bot \<le> fd + bot) x"
all_pos: shows "AE x in M. \<forall>i. 0 \<le> f i x"
O_in_G: fixes i :: "nat" shows "\<Union> (Q' ` {..i}) \<in> {Q \<in> sets M. emeasure N Q \<noteq> \<infinity>}"
list_assoc_those_compile_env_entries: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'k \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and xs :: "('k \<times> ('b, 'l, 'fun, 'd, 'e) Inca.instr fundef) list"   and ys :: "('k \<times> ('b, 'l, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef) list"   and f :: "'k" assumes "Result.those (map (compile_env_entry \<A> \<O>) xs) = Ok ys" shows "rel_option (\<lambda>fd1 fd2. compile_fundef \<A> (\<O> f) fd1 = Ok fd2) (map_of xs f) (map_of ys f)"
ball_imageD: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>f ` A. P x" shows "\<forall>x\<in>A. P (f x)"
tl_append: fixes xs :: "'a list"   and ys :: "'a list" shows "tl (xs @ ys) = (case xs of [] \<Rightarrow> tl ys | z # zs \<Rightarrow> zs @ ys)"
game\<^sub>0_game\<^sub>1: shows "run_gpv oracle game\<^sub>0 empty = map_spmf fst (sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> game\<^sub>1 x))"
cmod_char_le_1: fixes M :: "real measure"   and t :: "real" assumes "real_distribution M" shows "cmod (char M t) \<le> 1"
assms_1: shows "nsqn\<^sub>r r = nsqn\<^sub>r r'"
vec_nth: fixes x :: "('b, 'a) Finite_Cartesian_Product.vec" shows "($h) x \<in> UNIV"
vlrestriction_vdomain: shows "A \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> A = A"
bounded_byD: fixes b :: "real"   and P :: "'b \<Rightarrow> real"   and x :: "'b" assumes "bounded_by b P" shows "P x \<le> b"
trnr\<^sub>\<eta>_def: fixes v :: "'a"   and \<mu> :: "'a" shows "T0.trnr\<^sub>\<eta> v \<mu> \<equiv> (\<mu> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, f\<^sup>*, f] \<cdot> (v \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
simps: fixes st :: "nat"   and m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_inv st m lm (l, r) = (if st = 0 then wprepare_stop m lm (l, r) else if st = Suc 0 then wprepare_add_one m lm (l, r) else if st = Suc (Suc 0) then wprepare_goto_first_end m lm (l, r) else if st = Suc (Suc (Suc 0)) then wprepare_erase m lm (l, r) else if st = 4 then wprepare_goto_start_pos m lm (l, r) else if st = 5 then wprepare_loop_start m lm (l, r) else if st = 6 then wprepare_loop_goon m lm (l, r) else if st = 7 then wprepare_add_one2 m lm (l, r) else False)"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.cod f1', f2))"
map_key_PP_plus: fixes p :: "('d, 'e) pp \<Rightarrow>\<^sub>0 'f"   and q :: "('d, 'e) pp \<Rightarrow>\<^sub>0 'f" shows "Poly_Mapping.map_key PP (p + q) = Poly_Mapping.map_key PP p + Poly_Mapping.map_key PP q"
degree_monom: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and y :: "'a"   and v :: "nat" shows "MPoly_Type.degree (MPoly_Type.monom x y) v = (if y = (0::'a) then 0 else lookup x v)"
fmap_lifted_simps_1: fixes f :: "'b \<rightarrow> 'a" shows "fmap\<cdot>f\<cdot>\<bottom> = \<bottom>"
prenum_def: fixes e :: "nat"   and x :: "nat" shows "prenum e x \<equiv> Some (if x < e_length e then e_nth e x else 0)"
eq_self_neq_FF: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a" assumes "class.Eq_equiv eq" shows "eq\<cdot>x\<cdot>x \<noteq> FF"
rev_below_trans: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and z :: "'d"   and x :: "'d" assumes "class.po below"   and "below y z"   and "below x y" shows "below x z"
inj_on_map_upd: fixes m :: "'a \<Rightarrow> 'b option"   and y :: "'b"   and x :: "'a" assumes "inj_on m (dom m)"   and "Some y \<notin> range m" shows "inj_on (m(x \<mapsto> y)) (dom m)"
indRelRT_to_TRel: fixes TP :: "'procT"   and TQ :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm TQ" shows "(TP, TQ) \<in> TRel"
del_ops_distinct: fixes rects :: "'a rectangle list" shows "distinct (del_ops rects)"
language_def: shows "language \<equiv> {w. (!!) w \<Turnstile>\<^sub>p \<phi>}"
num: shows "(let (a, b) = quotient_of (ma_rat ma); (sq, fact) = prime_product_factor (nat (\<bar>a\<bar> * b)); ma' = ma_of_rat (rat_of_int (sgn a) * rat_of_nat sq / rat_of_int b) in ma_times ma' (ma_sqrt (ma_of_rat (rat_of_nat fact)))) = ma_times ma' sqrt"
inner_eucl_of_list: fixes i :: "'a"   and xs :: "real list" assumes "i \<in> Basis"   and "length xs = DIM('a)" shows "i \<bullet> eucl_of_list xs = xs ! index Basis_list i"
Chains_restrict_relD: fixes M :: "'b set"   and A :: "'b set"   and leq :: "('b \<times> 'b) set" assumes "M \<in> Chains (restrict_rel A leq)" shows "M \<in> Chains leq"
f_join_drop: fixes n :: "nat"   and xs :: "'b list"   and I :: "nat set" shows "xs \<up> n \<Join>\<^sub> I = xs \<Join>\<^sub> (I \<oplus> n)"
emeasure_null_measure: fixes M :: "'b measure"   and X :: "'b set" shows "emeasure (null_measure M) X = 0"
RF_subset_N: fixes i :: "nat"   and j :: "nat" assumes "i < m" shows "RF j i \<subseteq> N"
must_sync_def2: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong>1 = (\<exists>taa x' m' s. r1.r_syntax t x m taa x' m' \<and> r1.actions_ok s t taa)"
language_def: fixes \<phi> :: "'b pltl" shows "formula.language \<phi> \<equiv> {w. (!!) w \<Turnstile>\<^sub>p \<phi>}"
eq: shows "topspace X = {x \<in> topspace X. f x \<in> topspace Y}"
le: fixes X' :: "'v set" assumes "X' \<subseteq> X" shows "integral\<^sup>N (count_space X') f \<le> integral\<^sup>N (count_space (\<^bold>E `` X')) g"
sets_pre_sigmaI: fixes T :: "'a \<Rightarrow> 't"   and A :: "'a set" assumes "stopping_time F T"   and "\<And>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)" shows "A \<in> sets (pre_sigma T)"
pre_j: shows "prerecording_event t (Min {j \<in> {Max {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}} + 1..<length t}. prerecording_event t j})"
adv_P2_def: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool"   and D :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "etp.OT_12.adv_P2 n m1 m2 D = \<bar>spmf (etp.R2 n m1 m2 \<bind> D) True - spmf (funct_OT_12 m1 m2 \<bind> (\<lambda>(o1, o2). etp.S2 n m2 o2 \<bind> D)) True\<bar>"
exhaust: fixes y :: "meth_sig" obtains x1 :: "cl" and x2 :: "char list" and x3 :: "vd list" where "y = meth_sig_def x1 x2 x3"
disjCases2: fixes A :: "bool"   and B :: "bool" assumes "A \<or> B" obtains "A"   | "B"
Collect_triv: fixes a :: "'g"   and P :: "'g \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"
SeqSwap: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and R :: "'e set"   and c2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and c1 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c2 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c1 R,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Seq c1 c2 Q,A"
inj_on_Int: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f A \<or> inj_on f B" shows "inj_on f (A \<inter> B)"