hcju/isabelleps huggingface dataset:
subst_bvs1'_step: fixes a :: "Core.term"   and args :: "Core.term list"   and t :: "Core.term"   and lev :: "nat" assumes "\<forall>x\<in>set (a # args). is_closed x" shows "subst_bvs1' t lev (a # args) = subst_bvs1' (subst_bv2 t lev a) lev args"
rep_eq: fixes x :: "('c, 'd) fmap" shows "fmdom' x = dom (fmlookup x)"
abc_abd_acdadc: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "MinkowskiSpacetime \<E> \<P> betw"   and "betw a b c"   and "betw a b d"   and "c \<noteq> d" shows "betw a c d \<or> betw a d c"
fst_snd_flip: fixes xy :: "'c \<times> 'd" shows "fst xy = (snd \<circ> (\<lambda>(x, y). (y, x))) xy"
ide_apex: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "Limit.cone J C D a \<chi>" shows "partial_magma.ide C a"
collect_pair_finite: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "finite {x. P x}"   and "finite {x. Q x}" shows "finite {(x, y). P x \<and> Q y \<and> R x y}"
SeqSubstFormP_Mem: fixes s :: "name"   and k :: "name"   and x :: "tm"   and y :: "tm"   and x' :: "tm"   and y' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, x, y, x', y', v, i)"   and "atom k \<sharp> (x, y, x', y', v, i)" shows "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqSubstFormP v i (Q_Mem x y) (Q_Mem x' y') (Var s) (Var k)))"
hindrance_mono_web: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "hindrance \<Gamma>' f" shows "hindrance \<Gamma> f"
simps_3: fixes y :: "'d" shows "flat_orD (deny (allow y)) = deny y"
map_M_pure_E2: fixes h :: "'a"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and xs :: "'d list"   and ys :: "'c list"   and y :: "'c" assumes "h \<turnstile> map_M g xs \<rightarrow>\<^sub>r ys"   and "y \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (g x) h" obtains x :: "'d" where "h \<turnstile> g x \<rightarrow>\<^sub>r y" and "x \<in> set xs"
alg_dvd_mult: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a alg_dvd c"   and "algebraic_int b" shows "a alg_dvd b * c"
card_face_cycle_sets_da: shows "card H.face_cycle_sets = card G.face_cycle_sets"
hom_component_uminus: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "hom_component (- p) = - hom_component p"
contains_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and bot :: "'e"   and s :: "'e"   and r :: "'f \<times> 'g" shows "pre_RDR.contains \<delta> bot s r \<equiv> \<exists>rs. r \<in> set rs \<and> s = pre_RDR.exec \<delta> bot rs"
mem: shows "k \<in> set (concat (map (\<lambda>i. if v $ i \<noteq> (0::'a) then [i] else []) [0..<n]))"
FUnion_def: fixes xs :: "'b fset fset" shows "\<Union>\<^sub>fxs = Abs_fset (\<Union> (fset ` \<langle>xs\<rangle>\<^sub>f))"
r_inv_ex: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"
\<ll>'_ide_simp: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.\<ll>' a = \<F>C.lunit' a"
minus_eq: fixes x :: "complex"   and y :: "complex" shows "x + - y = x - y"
diamond_E: fixes r :: "('b \<times> 'b) set" assumes "\<diamond> r" obtains "r\<inverse> O r \<subseteq> r O r\<inverse>"
trans: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b"   and "b \<subseteq># c" shows "a \<subseteq># c"
assms: shows "\<forall>i j. mod_type_class.to_nat j \<le> k \<and> A $ i $ j \<noteq> (0::'b) \<longrightarrow> i \<le> j"
get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def: fixes character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap" shows "get character_data_ptr h = Option.bind (get (cast character_data_ptr) h) cast"
G_transpose_trancl: fixes A :: "(('b, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "(fixed_mat.G (Finite_Cartesian_Product.transpose A))\<^sup>+ = ((fixed_mat.G A)\<^sup>+)\<inverse>"
properfactor_trans1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a divides b"   and "properfactor G b c"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G a c"
Un_empty_right: fixes A :: "'a set" shows "A \<union> {} = A"
oosn_rreq: fixes i :: "nat" shows "opaodv i \<Turnstile> (otherwith quality_increases {i} (orecvmsg msg_fresh), other quality_increases {i} \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (seql i (\<lambda>(\<xi>, l). l \<in> {PAodv-:4, PAodv-:5} \<union> {PRreq-:n |n. True} \<longrightarrow> 1 \<le> osn \<xi>))"
powr_nat_bounds_ge_neg_1: fixes x :: "real"   and y :: "real"   and u :: "real" assumes "powr_nat \<bar>x\<bar> y \<le> u" shows "- u \<le> powr_nat x y"
proves_strip_all_single_Fv: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and \<tau> :: "typ"   and x :: "variable" assumes "\<Theta>,\<Gamma> \<turnstile> B"   and "strip_all_single_var B = Some \<tau>" shows "\<Theta>,\<Gamma> \<turnstile> subst_bv (Fv x \<tau>) (strip_all_single_body B)"
invertible_mat_def: fixes A :: "'a mat" shows "invertible_mat A \<equiv> square_mat A \<and> (\<exists>B. inverts_mat A B \<and> inverts_mat B A)"
Im2_def: fixes quat :: "quat" shows "Im2 quat = (case quat of Quat x1 x2 x3 x4 \<Rightarrow> x3)"
myUNIV_def: fixes x :: "'a"   and y :: "'a" shows "myUNIV x y = Star (Plus (Atom x) (Atom y))"
prob0_jozsa_algo_of_const_0: assumes "const 0" shows "prob0_fst_qubits n jozsa_algo = 1"
cases: fixes a1 :: "'e"   and a2 :: "'e" assumes "(a1, a2) \<in> hull_rel" obtains p :: "perm" and x :: "'e" where "a1 = p \<bullet> x" and "a2 = x"
zero_rev_flow_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> E" shows "f (v, u) = (0::'a)"
Amortized_def: fixes arity :: "'a \<Rightarrow> nat"   and exec :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b"   and inv :: "'b \<Rightarrow> bool"   and cost :: "'a \<Rightarrow> 'b list \<Rightarrow> nat"   and \<Phi> :: "'b \<Rightarrow> real"   and U :: "'a \<Rightarrow> 'b list \<Rightarrow> real" shows "Amortized arity exec inv cost \<Phi> U \<equiv> (\<forall>ss f. (\<forall>s\<in>set ss. inv s) \<longrightarrow> length ss = arity f \<longrightarrow> inv (exec f ss)) \<and> (\<forall>s. inv s \<longrightarrow> 0 \<le> \<Phi> s) \<and> (\<forall>ss f. (\<forall>s\<in>set ss. inv s) \<longrightarrow> length ss = arity f \<longrightarrow> real (cost f ss) + \<Phi> (exec f ss) - sum_list (map \<Phi> ss) \<le> U f ss)"
combine_prepend_replicate: fixes Ks :: "'a list list"   and Us :: "'a list list"   and n :: "nat" assumes "set Ks \<subseteq> carrier (K [X])"   and "set Us \<subseteq> carrier (K [X])" shows "UP.combine (replicate n \<zero>\<^bsub>K [X]\<^esub> @ Ks) Us = UP.combine Ks (drop n Us)"
X'_def: shows "X' \<equiv> (\<lambda>\<sigma>. \<sigma> x) -` X \<inter> space (state_measure V \<Gamma>)"
rv1_Cons: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and xa :: "bd_type_F + bd_type_F"   and xs :: "(bd_type_F + bd_type_F) list"   and x :: "'d" shows "rv1 s1 s2 (xa # xs) x = (case xa of Inl k1 \<Rightarrow> rv1 s1 s2 xs (frombd_F12 s1 x k1) | Inr k2 \<Rightarrow> rv2 s1 s2 xs (frombd_F13 s1 x k2))"
delete_def: fixes x :: "'c"   and t :: "(('c \<times> 'd) \<times> color) tree" shows "RBT_Map.delete x t = paint Black (RBT_Map.del x t)"
vrat_le_def: shows "vrat_le = ZFC_in_HOL.set {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> rat_of_vrat a \<le> rat_of_vrat b}"
simps_2: fixes verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and more :: "'d" shows "E\<^bsub>\<lparr>Graph.verts = verts, arcs = arcs, \<dots> = more\<rparr>\<^esub> = arcs"
simps_9: fixes f :: "'d \<Rightarrow> 'c"   and x21 :: "'d tree"   and x22 :: "'d"   and x23 :: "'d tree" shows "map_tree f \<langle>x21, x22, x23\<rangle> = \<langle>map_tree f x21, f x22, map_tree f x23\<rangle>"
rep_eq: fixes x :: "'e \<Rightarrow> 'd \<Rightarrow> real" shows "blinfun_apply (blinfun_of_matrix x) = (\<lambda>xa. \<Sum>i\<in>Basis. \<Sum>j\<in>Basis. (xa \<bullet> j * x i j) *\<^sub>R i)"
raw_match'_dom_res_eq_tvsT: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'" shows "dom subs' = tvsT T \<union> dom subs"
sum_zero_eq_neg: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "x \<oplus>\<^bsub>P\<^esub> y = \<zero>\<^bsub>P\<^esub>" shows "x = \<ominus>\<^bsub>P\<^esub> y"
lvars_pivot: shows "lvars (\<T> (pivot' x\<^sub>i x\<^sub>j s)) = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}"
sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> real" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> R" shows "(\<lambda>x. \<Sum>i\<in>I. f i x) \<in> R"
distinct_1: fixes x21 :: "'d"   and x22 :: "('d, 'e) Term.term list"   and x23 :: "('d, 'e) ctxt"   and x24 :: "('d, 'e) Term.term list" shows "\<box> \<noteq> More x21 x22 x23 x24"
weight_less_def: fixes s :: "('f, 'g) Term.term"   and t :: "('f, 'g) Term.term" shows "weight_less s t \<equiv> vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t) \<and> weight s < weight t"
elims_1: fixes x :: "atom fm"   and xa :: "real list"   and y :: "bool" assumes "eval x xa = y" obtains a :: "atom" and \<Gamma> :: "real list" where "x = fm.Atom a" and "xa = \<Gamma>__" and "y = aEval a \<Gamma>__"   | uu_ :: "real list" where "x = TrueF" and "xa = uu_" and "y"   | uv_ :: "real list" where "x = FalseF" and "xa = uv_" and "\<not> y"   | \<phi> :: "atom fm" and \<psi> :: "atom fm" and \<Gamma> :: "real list" where "x = And \<phi>__ \<psi>__" and "xa = \<Gamma>'__" and "y = (eval \<phi>__ \<Gamma>'__ \<and> eval \<psi>__ \<Gamma>'__)"   | \<phi> :: "atom fm" and \<psi> :: "atom fm" and \<Gamma> :: "real list" where "x = Or \<phi>'__ \<psi>" and "xa = \<Gamma>''__" and "y = (eval \<phi>'__ \<Gamma>''__ \<or> eval \<psi> \<Gamma>''__)"   | \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = Neg \<phi>''__" and "xa = \<Gamma>'''__" and "y = (\<not> eval \<phi>''__ \<Gamma>'''__)"   | \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = ExQ \<phi>'''__" and "xa = \<Gamma>''''__" and "y = (\<exists>x. eval \<phi>'''__ (x # \<Gamma>''''__))"   | \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = AllQ \<phi>''''__" and "xa = \<Gamma>'''''__" and "y = (\<forall>x. eval \<phi>''''__ (x # \<Gamma>'''''__))"   | i :: "nat" and \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = AllN i__ \<phi>'''''__" and "xa = \<Gamma>''''''__" and "y = (\<forall>l. length l = i__ \<longrightarrow> eval \<phi>'''''__ (l @ \<Gamma>''''''__))"   | i :: "nat" and \<phi> :: "atom fm" and \<Gamma> :: "real list" where "x = ExN i \<phi>" and "xa = \<Gamma>" and "y = (\<exists>l. length l = i \<and> eval \<phi> (l @ \<Gamma>))"
mk_eqcl_len_fst: fixes xs :: "nat option list"   and zs :: "nat list"   and i :: "nat"   and T :: "bool list list" shows "length (fst (mk_eqcl xs zs i T)) = length xs"
proj2_rep_dependent: fixes i :: "real"   and v :: "proj2"   and j :: "real"   and w :: "proj2" assumes "i *\<^sub>R proj2_rep v + j *\<^sub>R proj2_rep w = 0"   and "i \<noteq> 0 \<or> j \<noteq> 0" shows "v = w"
intro: fixes S :: "'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 S' :: "'f set"   and is_open' :: "'f set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "continuous_map S is_open S' is_open' f" shows "im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f"
elims: fixes x :: "J.Arr"   and y :: "'c" assumes "local.map x = y" obtains "x = J.AA" and "y = C.dom f0"   | "x = J.BB" and "y = C.dom f1"   | "x = J.TT" and "y = C.cod f0"   | "x = J.AT" and "y = f0"   | "x = J.BT" and "y = f1"   | "x = J.Null" and "y = C.null"
enough_flushs_nth_update: fixes i :: "nat"   and \<D> :: "bool"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "\<not> \<D> \<longrightarrow> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "enough_flushs (ts\<^sub>s\<^sub>b[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
less_a_def: fixes a :: "'a"   and b :: "'a" shows "less_a a b = (lesseq_a a b \<and> \<not> lesseq_a b a)"
distinct_5: fixes x2 :: "num"   and x3 :: "num" shows "num.Bit0 x2 \<noteq> num.Bit1 x3"
bezout_iterate_not_zero: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and k :: "nat"   and n :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "A $ i $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "n < nrows A"   and "mod_type_class.to_nat i \<le> n"   and "is_bezout_ext bezout" shows "bezout_iterate A n i (mod_type_class.from_nat k) bezout $ i $ mod_type_class.from_nat k \<noteq> (0::'d)"
dup_simps_2: fixes a :: "'a" assumes "ide a" shows "local.dom \<d>[a] = a"
inject_4: fixes x51 :: "'a formula"   and x52 :: "'a formula"   and y51 :: "'a formula"   and y52 :: "'a formula" shows "(x51 \<^bold>\<or> x52 = y51 \<^bold>\<or> y52) = (x51 = y51 \<and> x52 = y52)"
if_doesnt_read': fixes x :: "'Var"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" assumes "x \<notin> bexp_vars e"   and "\<C>_vars x \<inter> bexp_vars e = {}" shows "doesnt_read_or_modify (Stmt.If e c\<^sub>1 c\<^sub>2) x"
reads_consistent_snoc_Write\<^sub>s\<^sub>b: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" assumes "reads_consistent pending_write \<O> m sb" shows "reads_consistent pending_write \<O> m (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W])"
power_series_holomorphic: fixes z :: "complex"   and r :: "real"   and a :: "nat \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n * (w - z) ^ n) sums f w" shows "f holomorphic_on ball z r"
Vfrom_vsubset_VPow_Vfrom: fixes A :: "V"   and i :: "V" assumes "Transset A" shows "Vfrom A i \<subseteq>\<^sub>\<circ> VPow (Vfrom A i)"
unity_root_cong: fixes m :: "int"   and n :: "int"   and k :: "nat" assumes "[m = n] (mod int k)" shows "unity_root k m = unity_root k n"
less_le: fixes x :: "'a list"   and y :: "'a list" shows "strict_prefix x y = (prefix x y \<and> x \<noteq> y)"
simps: fixes x :: "'f"   and xs :: "'f list" shows "last (x # xs) = (if xs = [] then x else last xs)"
neg_assert_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.neg_assert one times inf dual bot x = inf (times (dual x) bot) one"
vec_hom_zero: fixes n :: "nat" shows "vec_hom (0\<^sub>v n) = 0\<^sub>v n"
l_Gs: shows "enat (Suc l) < llength Gs"
cexpr_typing_marg_dens_cexpr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat"   and vs :: "nat list" assumes "\<Gamma> \<turnstile>\<^sub>c \<delta> : REAL" shows "case_nat (\<Gamma> x) \<Gamma> \<turnstile>\<^sub>c marg_dens_cexpr \<Gamma> vs x \<delta> : REAL"
fun_pair_eq_subst: fixes d :: "('fun, 'b) Term.term \<times> ('fun, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('fun, 'a) Term.term"   and d' :: "('fun, 'c) Term.term \<times> ('fun, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('fun, 'a) Term.term" shows "(pair d \<cdot> \<delta> = pair d' \<cdot> \<theta>) = (d \<cdot>\<^sub>p \<delta> = d' \<cdot>\<^sub>p \<theta>)"
uniformly_continuous_on_add: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "uniformly_continuous_on s f"   and "uniformly_continuous_on s g" shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
affine_hull_Inter: fixes S :: "'a set set" assumes "\<not> affine_dependent (\<Union> S)" shows "affine hull \<Inter> S = \<Inter> ((hull) affine ` S)"
B: fixes x :: "nat" shows "(\<lambda>xa. \<forall>u. u 0 = 0 \<longrightarrow> prec_st (prec_count x) rel_count u xa) = (\<lambda>xa. \<forall>k. count xa k \<le> x)"
Levy_Inversion_aux2: fixes a :: "real"   and b :: "real"   and t :: "real" assumes "a \<le> b"   and "t \<noteq> 0" shows "cmod ((iexp (t * b) - iexp (t * a)) / (\<i> * complex_of_real t)) \<le> b - a"
msubstlepos_nb: fixes t :: "tm"   and a :: "poly"   and r :: "tm"   and c :: "poly" assumes "tmbound0 t"   and "islin (Le (CNP 0 a r))" shows "bound0 (msubstlepos c t a r)"
product_lists_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a list"   and ys :: "'a list list" shows "(x # xs \<in> set (product_lists (y # ys))) = (xs \<in> set (product_lists ys) \<and> x \<in> set y)"
cases: fixes a :: "trm" assumes "SNa a" obtains M :: "trm" where "a = M" and "\<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> SNa M'"
DirProd_monoid: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "Group.monoid G"   and "Group.monoid H" shows "Group.monoid (G \<times>\<times> H)"
Y_def: fixes f :: "'c" shows "Y f \<equiv> \<chi>.\<Phi>.Ya.Cop_S.Map (local.map f)"
create_gba_def: fixes \<phi> :: "'a ltlr" shows "create_gba \<phi> \<equiv> create_graph\<^sub>T \<phi> \<bind> (\<lambda>nds. RETURN (create_gba_from_nodes \<phi> nds))"
sumPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs" assumes "P \<leadsto><Rel> Q"   and "Rel \<subseteq> Rel'"   and "Id \<subseteq> Rel'" shows "P \<oplus> R \<leadsto><Rel'> Q \<oplus> R"
exhaust: fixes y :: "msg" obtains x11 :: "nat" and x12 :: "nat" and x13 :: "nat" and x14 :: "k" and x15 :: "nat" and x16 :: "nat" and x17 :: "nat" where "y = Rreq x11 x12 x13 x14 x15 x16 x17"   | x21 :: "nat" and x22 :: "nat" and x23 :: "nat" and x24 :: "nat" and x25 :: "nat" where "y = Rrep x21 x22 x23 x24 x25"   | x31 :: "nat \<Rightarrow> nat option" and x32 :: "nat" where "y = Rerr x31 x32"   | x41 :: "nat" and x42 :: "nat" where "y = Newpkt x41 x42"   | x51 :: "nat" and x52 :: "nat" and x53 :: "nat" where "y = Pkt x51 x52 x53"
ipresVar_def: fixes h :: "('c, 'd, 'varSort, 'var, 'e) qTerm set \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'i, 'j, 'var, 'f, 'k, 'l) Iteration.model_scheme" shows "ipresVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = igVar MOD xs x"
has_integral_spike_set_eq: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "negligible {x \<in> S - T. f x \<noteq> (0::'b)}"   and "negligible {x \<in> T - S. f x \<noteq> (0::'b)}" shows "(f has_integral y) S = (f has_integral y) T"
CompDiag_assoc: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "Diag t"   and "Diag u"   and "Diag v"   and "Dom t = Cod u"   and "Dom u = Cod v" shows "(t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v"
homeomorphic_minimal: fixes s :: "'b set"   and t :: "'c set" shows "s homeomorphic t = (\<exists>f g. (\<forall>x\<in>s. f x \<in> t \<and> g (f x) = x) \<and> (\<forall>y\<in>t. g y \<in> s \<and> f (g y) = y) \<and> continuous_on s f \<and> continuous_on t g)"
swap_id_eq: fixes a :: "'c"   and b :: "'c"   and x :: "'c" shows "(a \<rightleftharpoons>\<^sub>F b) x = (if x = a then b else if x = b then a else x)"
assms: shows "0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}"
vimage2p_Grp: fixes f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and P :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "BNF_Def.vimage2p f g P = BNF_Def.Grp UNIV f OO P OO (BNF_Def.Grp UNIV g)\<inverse>\<inverse>"
aboveS_subset_above: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "aboveS r a \<subseteq> above r a"
trms_of_deg_leq_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "trms_of_deg_leq n f \<equiv> f \<ominus>\<^bsub>UP R\<^esub> shift (Suc n) f \<otimes>\<^bsub>UP R\<^esub> monom P \<one> (Suc n)"
lunit_hcomp_2: fixes f :: "'d"   and g :: "'d" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D f, f, g] \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g] = \<l>\<^sub>D\<^sup>-\<^sup>1[f] \<star>\<^sub>D g"
cop_correct_ret: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and n :: "nat"   and i :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "path \<sigma> n = local.return"   and "obs \<sigma> i \<noteq> obs \<sigma>' i"   and "obs \<sigma>' i \<noteq> None"   and "\<sigma> =\<^sub>L \<sigma>'" shows "\<exists>k k'. ((\<sigma>, k), \<sigma>', k') \<in> cop \<or> ((\<sigma>', k'), \<sigma>, k) \<in> cop"
split_list_propE: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'b list" and x :: "'b" and zs :: "'b list" where "xs = ys @ x # zs" and "P x"
dg_prod_def: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "dg_prod I \<AA> = [\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Obj\<rparr>, \<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Arr\<rparr>, \<lambda>f\<in>\<^sub>\<circ>\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Arr\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Dom\<rparr>\<lparr>f\<lparr>i\<rparr>\<rparr>, \<lambda>f\<in>\<^sub>\<circ>\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Arr\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Cod\<rparr>\<lparr>f\<lparr>i\<rparr>\<rparr>]\<^sub>\<circ>"
dimension_independent: fixes K :: "'c set"   and Us :: "'c list" assumes "S.independent K Us" shows "S.dimension (length Us) K (S.Span K Us)"
simps_1: fixes r :: "'b rexp" shows "rderivs [] r = r"
field_inv_inv: fixes x :: "'a" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "x\<^bsup>\<hyphen> K\<^esup>\<^bsup>\<hyphen> K\<^esup> = x"