hcju/isabelleps huggingface dataset:
continuous_map_product_then_coordinatewise_1: fixes T1 :: "'b topology"   and T :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and i :: "'c" assumes "continuous_map T1 (product_topology T I) f"   and "i \<in> I" shows "continuous_map T1 (T i) (\<lambda>x. f x i)"
sup_idem: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) x = x"
valid_path_split_2: fixes as :: "'edge list"   and as' :: "'edge list" assumes "valid_path (as @ as')" shows "valid_path as'"
element_type_det: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "element_type \<tau> \<sigma>\<^sub>1"   and "element_type \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
chain_lemma: fixes S :: "nat \<Rightarrow> 'b"   and i :: "nat"   and k :: "nat" assumes "chain S" shows "S i \<sqsubseteq> S k \<or> S k \<sqsubseteq> S i"
uniqAppend_2: fixes l1 :: "'b list"   and l2 :: "'b list" assumes "uniq (l1 @ l2)" shows "uniq l2"
False1: shows "\<not> (b < a \<or> p = 0)"
B: shows "(x l\<rightarrow> (0::'a)) r\<rightarrow> -l x = (1::'a)"
rel_prod_sel: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'a \<times> 'c"   and q :: "'b \<times> 'd" shows "rel_prod R1 R2 p q = (R1 (fst p) (fst q) \<and> R2 (snd p) (snd q))"
prefixes_stable: fixes t :: "nat"   and j :: "nat"   and b :: "nat"   and d :: "nat"   and b' :: "nat" assumes "local.prefixes t j \<down>= b"   and "local.prefixes (t + d) j \<down>= b'" shows "\<forall>x<e_length b. e_nth b x = e_nth b' x"
imageE: fixes b :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "b \<in> f ` A" obtains x :: "'d" where "b = f x" and "x \<in> A"
simps_1: fixes uu :: "agent" shows "catomic (cAgent uu) = True"
infiter_inf_top: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<infinity> = c\<^sup>\<omega> ; \<top>"
abs_rep2_ID: fixes pat :: "[:'r:]" shows "abs2 oo rep2\<cdot>pat = ID"
left_inverse_linear: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "linear (*s) (*s) f"   and "g \<circ> f = id" shows "linear (*s) (*s) g"
le_supE: fixes a :: "'c"   and b :: "'c"   and x :: "'c" assumes "a + b \<le> x" obtains "a \<le> x" and "b \<le> x"
tendsto_unique: fixes "open" :: "'e set \<Rightarrow> bool"   and F :: "'f filter"   and f :: "'f \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.t2_space open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open a) F"   and "filterlim f (topological_space.nhds open b) F" shows "a = b"
integral_f: fixes n :: "nat" shows "integral {a..} (f n) = (if a \<le> real n then F (real n) - F a else 0)"
add_carrier_mat: fixes B :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and A :: "'a mat" assumes "B \<in> carrier_mat nr nc" shows "A + B \<in> carrier_mat nr nc"
may_acquire_all_has_locks_acquire_locks: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and ln :: "'a \<Rightarrow>f nat"   and t' :: "'b"   and l :: "'a" assumes "may_acquire_all ls t ln"   and "t \<noteq> t'" shows "has_locks (acquire_locks (ls $ l) t (ln $ l)) t' = has_locks (ls $ l) t'"
antisym: fixes b :: "'a multiset"   and a :: "'a multiset" assumes "b \<subseteq># a"   and "a \<subseteq># b" shows "a = b"
mod_subset_def: fixes p :: "nat"   and X :: "'c topology" shows "mod_subset p X \<equiv> {(a, b). singular_chain p X (a - b)}"
that: fixes k :: "'a" assumes "k \<in> fst ` set (fst (list_of_oalistx xs)) \<union> fst ` set (fst (list_of_oalisty ys))"   and "aux = f k (lookup xs k) (oay.lookup ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set (fst (list_of_oalistx xs)) \<union> fst ` set (fst (list_of_oalisty ys)); lt ko k' k\<rbrakk> \<Longrightarrow> f k' (lookup xs k') (oay.lookup ys k') = Some Eq" shows "thesis"
fds_scale_of_nat: fixes c :: "nat"   and n :: "nat" assumes "0 < c" shows "fds_scale c (of_nat n) = of_nat n"
le_sup_iff: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "less_eq (sup x y) z = (less_eq x z \<and> less_eq y z)"
all_zero_below_greatest: shows "\<forall>a>GREATEST m. \<not> is_zero_row_upt_k m k A. is_zero_row_upt_k a (Suc k) A"
assms_4: shows "\<guillemotleft>\<eta> : src f \<Rightarrow> g \<star> f\<guillemotright>"
acc_modi_top1: fixes a :: "acc_modi" shows "a \<le> Public"
var_swap_ref2: fixes a :: "'b"   and b :: "'b" shows "(''x'' ::= (\<lambda>s. s ''y'')) ; rel_R \<lceil>\<lambda>s. s ''z'' = a \<and> s ''x'' = b\<rceil> \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil> \<subseteq> rel_R \<lceil>\<lambda>s. s ''z'' = a \<and> s ''y'' = b\<rceil> \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil>"
affine_translation: fixes S :: "'c set"   and a :: "'c" shows "affine S = affine ((+) a ` S)"
frac_eq: fixes x :: "'b" shows "(frac x = x) = ((0::'b) \<le> x \<and> x < (1::'b))"
assms_2: shows "freeword_funlift id c = (0::'w)"
axioms: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" assumes "morphism_sheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
surj_imp_inj_inv: fixes f :: "'c \<Rightarrow> 'b" assumes "top_pres ((`) f)" shows "inj (inv f)"
DBM_reset'_sound_empty: fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and cs :: "'c list"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and u :: "'c \<Rightarrow> 'd" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n"   and "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d" shows "\<not> u \<turnstile>\<^bsub>v,n\<^esub> M"
unlabel_def: fixes S :: "('b \<times> 'a) list" shows "unlabel S \<equiv> map snd S"
simps: fixes CC :: "'d set set"   and F :: "'d program"   and a1 :: "'d set"   and a2 :: "'d set" shows "eltp CC F a1 a2 = ((\<exists>A B. a1 = A \<and> a2 = B \<and> F \<in> A ensures B \<and> A - B \<in> insert {x. bot x} CC) \<or> (\<exists>A B C. a1 = A \<and> a2 = C \<and> eltp CC F A B \<and> eltp CC F B C) \<or> (\<exists>S B. a1 = \<Union> S \<and> a2 = B \<and> (\<forall>A\<in>S. eltp CC F A B)))"
algebraic_altdef_ipoly: fixes x :: "'a" shows "algebraic x = (\<exists>p. ipoly p x = (0::'a) \<and> p \<noteq> 0)"
lset_lnth_member: fixes xs :: "'b llist"   and A :: "'b set"   and n :: "nat" assumes "lset xs \<subseteq> A"   and "enat n < llength xs" shows "xs $ n \<in> A"
redT_normal: fixes t :: "'t"   and x :: "'x1"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x' :: "'x1"   and m' :: "'m1"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "r1.r_syntax t x (shr s) ta x' m'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "r1.actions_ok s t ta"   and "redT_upd s t ta x' m' s'" shows "s -1-t\<triangleright>ta\<rightarrow> s'"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "inj_zero_hom hom" shows "map_poly_inj_zero_hom hom"
F_Un_D_card: shows "card (F \<union> D) = nat (int ((p - 1) div 2))"
inf_mono: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (inf a b) (inf c d)"
cont_left_sync: fixes Y :: "nat \<Rightarrow> 'c process"   and A :: "'c set"   and S :: "'c process" assumes "chain Y" shows "((\<Squnion>i. Y i) \<lbrakk>A\<rbrakk> S) = (\<Squnion>i. Y i \<lbrakk>A\<rbrakk> S)"
assms_3: shows "\<forall>x\<in>U\<^sub>3. f x \<in> U"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Can (t \<^bold>\<otimes> u) = (Can t \<and> Can u)"
impl: fixes s :: "'a" assumes "s \<in> P" shows "s \<in> P' s \<and> (\<forall>t. t \<in> Q' s \<longrightarrow> t \<in> Q) \<and> (\<forall>t. t \<in> A' s \<longrightarrow> t \<in> A)"
DERIV_chain': fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and E :: "'a" assumes "(f has_field_derivative D) (at x within s)"   and "(g has_field_derivative E) (at (f x))" shows "((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
Red_I_of_Red_F_subset: fixes N' :: "'f set"   and N :: "'f set" assumes "N' \<subseteq> Red_F N" shows "Red_I N \<subseteq> Red_I (N - N')"
equal_pdevs_def: fixes a :: "'d pdevs"   and b :: "'d pdevs" shows "equal_class.equal a b = (a = b)"
Seq_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and e2 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (e;; e2, xs) (e';; e2, xs')"
finite_comm_group_def: fixes G :: "('c, 'd) monoid_scheme" shows "finite_comm_group G \<equiv> comm_group G \<and> finite_comm_group_axioms G"
interaction_any_bounded_by_absorb: fixes gpv :: "('i, 'j) distinguisher"   and bound1 :: "enat"   and conv :: "('i, 'j, 'k, 'l) converter"   and bound2 :: "enat" assumes "interaction_any_bounded_by gpv bound1"   and "interaction_any_bounded_converter conv bound2" shows "interaction_any_bounded_by (absorb gpv conv) (bound1 * bound2)"
compact_imp_countably_compact: fixes U :: "'b set" assumes "compact U" shows "countably_compact U"
akra_bazzi_p_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "(\<Sum>i<k. as ! i * bs ! i powr y) \<le> (\<Sum>i<k. as ! i * bs ! i powr x)"
surjective: fixes r :: "'a transition_scheme" shows "r = \<lparr>Label = Label r, Arity = Arity r, Guards = Guards r, Outputs = Outputs r, Updates = Updates r, \<dots> = transition.more r\<rparr>"
local_time_unbounded3: fixes A :: "'a set"   and B :: "'a set"   and k :: "nat" assumes "A \<in> sets M"   and "B \<in> sets M"   and "emeasure M A < \<infinity>"   and "A - (\<Union>i. (T ^^ i) --` B) \<in> null_sets M" shows "(\<lambda>n. emeasure M {x \<in> (T ^^ n) --` A. local_time B n x < k}) \<longlonglongrightarrow> 0"
trans_not_greater: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b \<noteq> Greater"   and "cmp b c \<noteq> Greater" shows "cmp a c \<noteq> Greater"
VDM_valid_validn: fixes c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and n :: "nat" assumes " \<Turnstile> c : A " shows " \<Turnstile>\<^sub>n c : A "
that: fixes C :: "'a set" assumes "C \<in> components S"   and "bounded C" shows "thesis"
similarity_eye': fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "similarity (1, 0, 0, 1) M = M"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree" shows "consistent (Node w t\<^sub>1 t\<^sub>2) = (alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {} \<and> consistent t\<^sub>1 \<and> consistent t\<^sub>2)"
TOTALS: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "\<forall>a\<in>Field r. \<forall>b\<in>Field r. (a, b) \<in> r \<or> (b, a) \<in> r"
mult_zero_circ: fixes x :: "'a" shows "(x * bot)\<^sup>\<star> = Rf x bot"
expectation_gpv_parametric': shows "((=) ===> rel_\<I> C (=) ===> (A ===> (=)) ===> rel_gpv'' A C (=) ===> (=)) expectation_gpv expectation_gpv"
rs'_loopfree: shows "defNode g r \<notin> set (tl rs')"
rotate_is_Nil_conv: fixes n :: "nat"   and xs :: "'c list" shows "(rotate n xs = []) = (xs = [])"
TheEqI: fixes \<phi> :: "\<nu> \<Rightarrow> \<o>"   and \<psi> :: "\<nu> \<Rightarrow> \<o>" assumes "\<And>x. [\<phi> x in dw] = [\<psi> x in dw]" shows "(\<^bold>\<iota>x. \<phi> x) = (\<^bold>\<iota>x. \<psi> x)"
iso_runit: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b"   and f :: "'b" assumes "right_hom_with_unit V H \<a> \<iota> a"   and "partial_magma.ide (subcategory.comp V (weak_composition.right V H a)) f" shows "category.iso (subcategory.comp V (weak_composition.right V H a)) (right_hom_with_unit.runit V H \<a> \<iota> a f)"
deriv_bound_iff: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and n :: "nat" shows "deriv_bound r a n = (\<forall>b m. n < m \<longrightarrow> (a, b) \<notin> r ^^ m)"
assoc_aug_assoc: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "assoc_bpp (aug_pm_set z i {a}) f" shows "assoc_bpp {a} f"
rEqualsT: fixes \<theta>\<^sub>1 :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and r1 :: "real"   and \<theta>\<^sub>2 :: "trm"   and r2 :: "real" assumes "[\<theta>\<^sub>1]\<nu> \<down> r1"   and "[\<theta>\<^sub>2]\<nu> \<down> r2"   and "r1 = r2" shows "[Equals \<theta>\<^sub>1 \<theta>\<^sub>2]\<nu> \<downharpoonright> True"
i_take_eq_Nil: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "(f \<Down> n = []) = (n = 0)"
hmset_pos_plus: fixes A :: "zhmultiset"   and B :: "zhmultiset" shows "hmset_pos (A + B) = hmset_pos A - hmset_neg B + (hmset_pos B - hmset_neg A)"
iff: fixes X :: "'a set"   and Y :: "'b set" assumes "X homeomorphic Y" shows "P X = Q Y"
swap_commute: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "Fun.swap a b f = Fun.swap b a f"
simps_6: fixes t1 :: "PrimScheme"   and t2 :: "PrimScheme" shows "sch_to_nat (Pair_op t1 t2) = c_pair 5 (c_pair (sch_to_nat t1) (sch_to_nat t2))"
multlist_closed: fixes fs :: "'a vec list" assumes "set fs \<subseteq> carrier_vec n" shows "foldr (+) fs (0\<^sub>v n) \<in> carrier_vec n"
id: shows "snd ` set (pivot_positions A) = {0..<n}"
cases: fixes a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" assumes "ord_resolve_rename a1 a2 a3 a4 a5 a6" obtains CAs :: "'a literal multiset list" and n :: "nat" and AAs :: "'a multiset list" and As :: "'a list" and DA :: "'a literal multiset" and \<rho> :: "'s" and \<rho>s :: "'s list" and \<sigma> :: "'s" and E :: "'a literal multiset" where "a1 = CAs" and "a2 = DA" and "a3 = AAs" and "a4 = As" and "a5 = \<sigma>" and "a6 = E" and "length CAs = n" and "length AAs = n" and "length As = n" and "\<forall>i<n. poss (AAs ! i) \<subseteq># CAs ! i" and "negs (mset As) \<subseteq># DA" and "\<rho> = hd (renamings_apart (DA # CAs))" and "\<rho>s = tl (renamings_apart (DA # CAs))" and "ord_resolve (CAs \<cdot>\<cdot>cl \<rho>s) (DA \<cdot> \<rho>) (AAs \<cdot>\<cdot>aml \<rho>s) (As \<cdot>al \<rho>) \<sigma> E"
Quorum_not_empty: shows "\<exists>Q. Q \<in> Quorum"
extr_monotone_chan: fixes c :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and G :: "chan set"   and H :: "chan set" assumes "c \<in> extr bad IK G"   and "G \<subseteq> H" shows "c \<in> extr bad IK H"
non_distinct_duplicate_indices: fixes xs :: "'d list" assumes "\<not> distinct xs" shows "\<exists>i1 i2. i1 \<noteq> i2 \<and> xs ! i1 = xs ! i2 \<and> i1 \<le> length xs \<and> i2 \<le> length xs"
splitAt_distinct_fst: fixes vs :: "'c list"   and ram1 :: "'c" assumes "distinct vs" shows "distinct (fst (splitAt ram1 vs))"
edges_of_walk_def: fixes xs :: "'a list" shows "edges_of_walk xs = {uu_. \<exists>v w xs_pre xs_post. uu_ = (v, w) \<and> xs = xs_pre @ v # w # xs_post}"
deepL_list: fixes n :: "nat"   and m :: "('b, 'c) FingerTreeStruc"   and sf :: "('b, 'c) Digit"   and pr :: "('b, 'c) Node list" assumes "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"   and "is_leveln_digit n sf \<and> is_measured_digit sf"   and "\<forall>x\<in>set pr. (is_measured_node x \<and> is_leveln_node n x) \<and> length pr \<le> 4" shows "toList (deepL pr m sf) = nlistToList pr @ toList m @ digitToList sf"
\<phi>_naturality: fixes f :: "'a"   and x :: "'a"   and x' :: "'a"   and g :: "'b"   and y' :: "'b"   and y :: "'b"   and h :: "'a" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>A x'\<guillemotright>"   and "\<guillemotleft>g : y' \<rightarrow>\<^sub>B y\<guillemotright>"   and "\<guillemotleft>h : F y \<rightarrow>\<^sub>A x\<guillemotright>" shows "\<phi> y' (f \<cdot>\<^sub>A h \<cdot>\<^sub>A F g) = G f \<cdot>\<^sub>B \<phi> y h \<cdot>\<^sub>B g"
mset_ran_subst_inside: fixes r :: "int set"   and i :: "int"   and a :: "int \<Rightarrow> 'a"   and x :: "'a" assumes "finite r"   and "i \<in> r" shows "mset_ran (a(i := x)) r = mset_ran a r + {#x#} - {#a i#}"
USUP_all_cong: fixes F :: "'a \<Rightarrow> ('b, 'c) uexpr"   and G :: "'a \<Rightarrow> ('b, 'c) uexpr" assumes "\<And>P. F P = G P" shows "(\<Squnion> P \<bullet> F P) = (\<Squnion> P \<bullet> G P)"
psubsetD: fixes A :: "'d set"   and B :: "'d set"   and c :: "'d" assumes "A \<subset> B"   and "c \<in> A" shows "c \<in> B"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Arr \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Arr t \<and> Arr u \<and> Arr v \<and> Src t = Trg u \<and> Src u = Trg v)"
a: shows "\<exists>f. \<partial>\<^sub>F f \<stileturn> \<partial>\<^sub>F h"
hyps_3: shows "c' \<turnstile> ev \<mapsto> c''"
size_6: fixes x31 :: "'a"   and x32 :: "'a ifex"   and x33 :: "'a ifex" shows "size (IF x31 x32 x33) = size x32 + size x33 + Suc 0"
list_all_append: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
children_rank_less: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tree_invar t" shows "\<forall>t'\<in>set (SkewBinomialHeapStruc.children t). SkewBinomialHeapStruc.rank t' < SkewBinomialHeapStruc.rank t"
new_messages_keep_consistency: fixes M :: "('a \<times> nat, 'b) message set"   and i :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) message" assumes "consistent M"   and "event_at i (Send m)"   and "set (received_messages i) \<subseteq> M"   and "i \<notin> I ` insert_messages M" shows "consistent (insert m M)"
CBlinfun_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. bounded_clinear f}" shows "(*\<^sub>V) (CBlinfun y) = y"
compl: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "rational_preference carrier relation" shows "\<forall>x\<in>carrier. \<forall>y\<in>carrier. x \<succeq>[relation] y \<or> y \<succeq>[relation] x"
assms_6: shows "\<not> (\<exists>(a', b', c')\<in>set d. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
Node_1: fixes i :: "int"   and j :: "int" assumes "inorder l = [i..j]" shows "wpl i j l = Wpl i l"
interior_hyperplane: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)" shows "interior {x. a \<bullet> x = b} = {}"
DisjI1: fixes A :: "'a set"   and B :: "'a set"   and X :: "'a set" assumes "A \<inter> B = {}" shows "(A - X) \<inter> B = {}"
64271: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> (y \<squnion> - - (y \<sqinter> x))) = - (- x \<squnion> y)"
h2: shows " \<Gamma> \<subseteq> \<Gamma>' "
assms: shows "h \<turnstile> local.a_set_disconnected_nodes document_ptr disc_nodes \<rightarrow>\<^sub>h h'"
P: shows "padic_mult p f g (Suc (nat (padic_val p f + padic_val p g))) mod p ^ Suc (nat (padic_val p f + padic_val p g)) = f (Suc (nat (padic_val p f + padic_val p g))) * g (Suc (nat (padic_val p f + padic_val p g))) mod p ^ Suc (nat (padic_val p f + padic_val p g))"
lg_sym: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.QCong Cong l"   and "l A B" shows "l B A"
lemma_2_5_12_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c r\<rightarrow> a \<le> c r\<rightarrow> b"
isIdom_def: fixes g :: "'g"   and n :: "'node"   and m :: "'node" shows "isIdom g n m = ((m \<noteq> n \<and> dominates g m n) \<and> (\<forall>m'\<in>set (\<alpha>n g). m' \<noteq> n \<and> dominates g m' n \<longrightarrow> dominates g m' m))"
foundedD: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool"   and delta :: "'b" assumes "founded subs P (tree subs delta)"   and "terminal subs delta" shows "P delta"
add_mset_in_multiset: fixes M :: "'k \<Rightarrow> nat"   and a :: "'k" assumes "M \<in> multiset" shows "(\<lambda>b. if b = a then Suc (M b) else M b) \<in> multiset"
bind_def: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'g set" shows "Set.bind A f = {x. \<exists>B\<in>f ` A. x \<in> B}"