hcju/isabelleps huggingface dataset:
prim_vc_2: fixes minarc :: "'b \<Rightarrow> 'b"   and sum :: "'b \<Rightarrow> 'b"   and plus :: "'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 bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and t :: "'b"   and v :: "'b"   and g :: "'b"   and r :: "'b" assumes "class.m_kleene_algebra minarc sum plus inf less_eq less sup bot top uminus one times conv star"   and "m_kleene_algebra.prim_spanning_invariant inf less_eq top uminus one times conv star t v g r"   and "inf (times v (uminus (conv v))) g \<noteq> bot" shows "m_kleene_algebra.prim_spanning_invariant inf less_eq top uminus one times conv star (sup t (minarc (inf (times v (uminus (conv v))) g))) (sup v (times (conv (minarc (inf (times v (uminus (conv v))) g))) top)) g r \<and> card {x. x = uminus (uminus x) \<and> less_eq x (times (conv r) (star (uminus (uminus g)))) \<and> less_eq x (uminus (conv (sup v (times (conv (minarc (inf (times v (uminus (conv v))) g))) top))))} < card {x. x = uminus (uminus x) \<and> less_eq x (times (conv r) (star (uminus (uminus g)))) \<and> less_eq x (uminus (conv v))}"
sum_relI_1: fixes l :: "'c"   and l' :: "'d"   and Rl :: "('c \<times> 'd) set"   and Rr :: "('e \<times> 'f) set" assumes "(l, l') \<in> Rl" shows "(Inl l, Inl l') \<in> \<langle>Rl, Rr\<rangle>sum_rel"
that: fixes S :: "'a set" assumes "finite S"   and "cbox (x - sum ((*\<^sub>R) d) Basis) (x + sum ((*\<^sub>R) d) Basis) = convex hull S" shows "thesis"
surjective: fixes r :: "('d, 'e, 'f, 'g) slens_scheme" shows "r = \<lparr>view = \<V>\<^bsub>r\<^esub>, coview = \<C>\<^bsub>r\<^esub>, \<dots> = slens.more r\<rparr>"
injectivity: fixes x :: "'a"   and y :: "'a" assumes "to_fract x = to_fract y" shows "x = y"
PiM_cong: fixes I :: "'a set"   and J :: "'a set"   and M :: "'a \<Rightarrow> 'b measure"   and N :: "'a \<Rightarrow> 'b measure" assumes "I = J"   and "\<And>x. x \<in> I \<Longrightarrow> M x = N x" shows "Pi\<^sub>M I M = Pi\<^sub>M J N"
learn_cpI: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "environment \<psi> U s"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>i. \<psi> i = f \<and> (\<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i)"   and "\<And>f n. f \<in> U \<Longrightarrow> \<psi> (the (s (f \<triangleright> n))) \<in> U" shows "learn_cp \<psi> U s"
start_tid_def: shows "start_tid = addr2thread_id (hd start_addrs)"
case_prod_beta': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
assms_1: shows "algebraic_semidom_class.coprime p q"
integral_integrable_2dim: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "continuous_on (cbox (a, c) (b, d)) f" shows "(\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y))) integrable_on cbox a b"
head_arev: fixes a :: "'a \<times> 'a" assumes "a \<in> arcs (with_proj G)" shows "head (with_proj G) (edge_rev M a) = tail (with_proj G) a"
0: shows "\<guillemotleft>\<pp>\<^sub>0[a, b] : a \<otimes> b \<rightarrow> b\<guillemotright>"
prim_pyth_tripleI1: fixes v :: "nat"   and u :: "nat" assumes "0 < v"   and "v < u"   and "coprime u v"   and "\<not> (odd u \<and> odd v)" shows "prim_pyth_triple (2 * u * v) (u\<^sup>2 - v\<^sup>2) (u\<^sup>2 + v\<^sup>2)"
hd_exp_powser': fixes cs :: "real msstream"   and f :: "('a \<times> real) msllist" shows "ms_aux_hd_exp (powser_ms_aux' cs f) = Some 0"
chamber_vertices: fixes S :: "'b set"   and w :: "'b"   and s :: "'b" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>"   and "s \<in> S" shows "w +o \<langle>S - {s}\<rangle> \<in> dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w}"
elims: fixes x :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and xa :: "'b\<^sub>\<bottom>"   and xb :: "exp"   and y :: "exp" assumes "lift_transform x xa xb = y" obtains t :: "'b \<Rightarrow> exp \<Rightarrow> exp" and e :: "exp" where "x = t__" and "xa = Ibottom" and "xb = e__" and "y = e__"   | t :: "'b \<Rightarrow> exp \<Rightarrow> exp" and a :: "'b" and e :: "exp" where "x = t" and "xa = Iup a" and "xb = e" and "y = t a e"
semiring_normalization_rules_29: fixes x :: "'a" shows "x * x = x\<^sup>2"
wa_refine_def: fixes WAC :: "'a while_algo"   and WAA :: "'b while_algo"   and \<alpha> :: "'a \<Rightarrow> 'b" shows "wa_refine WAC WAA \<alpha> \<equiv> ((\<forall>s. s \<in> wa_invar WAC \<longrightarrow> s \<in> wa_cond WAC \<longrightarrow> \<alpha> s \<in> wa_cond WAA) \<and> (\<forall>s s'. s \<in> wa_invar WAC \<longrightarrow> s \<in> wa_cond WAC \<longrightarrow> (s, s') \<in> wa_step WAC \<longrightarrow> (\<alpha> s, \<alpha> s') \<in> wa_step WAA)) \<and> \<alpha> ` wa_initial WAC \<subseteq> wa_initial WAA \<and> \<alpha> ` wa_invar WAC \<subseteq> wa_invar WAA"
vector_minus_component: fixes x :: "('d, 'e) vec"   and y :: "('d, 'e) vec"   and i :: "'e" shows "(x - y) $ i = x $ i - y $ i"
distinct_3: fixes x1 :: "'b"   and x31 :: "'b local.term"   and x32 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> x31 \<^bold>\<otimes> x32"
step_nonneg': fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and i :: "nat"   and x :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "i < k"   and "x\<^sub>1 \<le> x" shows "0 \<le> bs ! i + (hs ! i) x / x"
assms_2: shows "Broadcast (i1, Insert e1 n1) \<in> set (history i)"
cong_mod_left: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "[b mod a = c] (mod a) = [b = c] (mod a)"
Fract_conv_to_fract: fixes a :: "'b"   and b :: "'b" shows "Fraction_Field.Fract a b = to_fract a / to_fract b"
is_picks_def: fixes dining_event :: "dining_event" shows "is_picks dining_event = (\<exists>x11 x12. dining_event = picks x11 x12)"
arr_MkArr: fixes A :: "'a"   and B :: "'a"   and f :: "'a set" assumes "A \<in> Collect obj"   and "B \<in> Collect obj"   and "f \<in> Maps.Hom A B" shows "Maps.arr (Maps.MkArr A B f)"
eval_wlp_Abort: fixes P :: "'a \<Rightarrow> real" shows "wlp Abort P = (\<lambda>s. 1)"
subrelI: fixes r :: "('a \<times> 'c) set"   and s :: "('a \<times> 'c) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
solutions_linorder_strict: fixes a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat" assumes "solution (a, b)"   and "solution (x, y)" shows "(a, b) = (x, y) \<or> a < x \<and> b < y \<or> x < a \<and> y < b"
simps_2: fixes x21 :: "LambdaType.type"   and x22 :: "LambdaType.type"   and y21 :: "LambdaType.type"   and y22 :: "LambdaType.type" shows "(x21 \<Rightarrow> x22 = y21 \<Rightarrow> y22) = (x21 = y21 \<and> x22 = y22)"
insert_image: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "x \<in> A" shows "insert (f x) (f ` A) = f ` A"
GroupHom_restrict0_subgroup: fixes H :: "'g set" assumes "Subgroup H" shows "GroupHom H (T \<down> H)"
assms_4: shows "D r \<upsilon> \<sigma>' \<sigma>'' \<upsilon>'"
update_convs_2: fixes yc' :: "int \<Rightarrow> int"   and xc :: "int"   and yc :: "int"   and more :: "'a" shows "yc_update yc' \<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = \<lparr>xc = xc, yc = yc' yc, \<dots> = more\<rparr>"
num_elems_of_ord_eq_phi': fixes d :: "nat" assumes "finite (carrier R)"   and "d dvd order (mult_of R)"   and "\<exists>a\<in>carrier (mult_of R). G.ord a = d" shows "card {a \<in> carrier (mult_of R). G.ord a = d} = phi' d"
k_def: shows "k \<equiv> SOME k. k \<noteq> 0"
assms_4: shows "\<forall>n. f (x n) \<le> c n"
depressed_quartic_Ferrari: fixes m :: "'a"   and p :: "'a"   and r :: "'a"   and q :: "'a"   and a :: "'a"   and b1 :: "'a"   and b2 :: "'a"   and y :: "'a" assumes "(8::'a) * m ^ 3 + (8::'a) * p * m\<^sup>2 + ((2::'a) * p\<^sup>2 - (8::'a) * r) * m - q\<^sup>2 = (0::'a)"   and "q \<noteq> (0::'a)"   and "a * a = (2::'a) * m"   and "b1 = p / (2::'a) + m - q / ((2::'a) * a)"   and "b2 = p / (2::'a) + m + q / ((2::'a) * a)" shows "(y ^ 4 + p * y\<^sup>2 + q * y + r = (0::'a)) = (poly [:b1, a, 1::'a:] y = (0::'a) \<or> poly [:b2, - a, 1::'a:] y = (0::'a))"
arcs_0_4: shows "f y0 \<in> arcs G"
assms_1: shows "set \<phi>1s \<subseteq> fmla"
edka_complexity_refine: shows "edka_complexity \<le> \<Down> Id edka"
bigo_minus: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(g)" shows "- f \<in> O(g)"
elims_3: fixes x :: "'a option"   and xa :: "'a option" assumes "\<not> x \<le> xa" obtains x :: "'a" where "xa__ = Some xb__" and "xa = None"   | x :: "'a" and y :: "'a" where "xa__ = Some x" and "xa = Some y" and "\<not> x \<le> y"
noDA: fixes xs :: "('a, 'b) Combinators list"   and s :: "('a, 'b) Combinators" assumes "noDenyAll xs"   and "s \<in> set xs" shows "\<not> member DenyAll s"
nn_integral_cmult: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. c * f x \<partial>M = c * integral\<^sup>N M f"
clop_idem_var: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "clop f" shows "f (f x) = f x"
0: shows "g [^]\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> n \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
assms_1: shows "\<forall>X\<in>XX. finite X"
monom_vars_eval_monom: fixes m :: "'c monom"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> monom_vars m \<Longrightarrow> f x = g x" shows "eval_monom f m = eval_monom g m"
transform1_rev: fixes P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<forall>\<tau>. \<tau> \<Turnstile> P" shows "P = true"
quot_of_fract_inverse: fixes x :: "'b fract" shows "quot_of_fract (inverse x) = (let (a, b) = quot_of_fract x; d = unit_factor a in if d = (0::'b) then (0::'b, 1::'b) else (b div d, a div d))"
assms: shows "(u, v) \<in> (g_E G')\<^sup>*"
analz_image_Key: fixes N :: "nat set" shows "analz (Key ` N) = Key ` N"
quasiconvexC: fixes C :: "real"   and X :: "'b set" assumes "quasiconvex C X" shows "0 \<le> C"
map_disc_iff: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b literal" shows "is_pos (map_literal f a) = is_pos a"
swap_Op_simp: fixes inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var"   and delta :: "'c" assumes "goodInp inp"   and "goodBinp binp" shows "Op delta inp binp #[x1 \<and> x2]_xs = Op delta (inp %[x1 \<and> x2]_xs) (binp %%[x1 \<and> x2]_xs)"
Ax_in_CANDs_1: fixes y :: "name"   and x :: "name"   and a :: "coname"   and B :: "ty" shows "y:Ax x a \<in> \<parallel>(B)\<parallel>"
prefix_tl_subset: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys" shows "set (tl xs) \<subseteq> set (tl ys)"
nat_less_eq_zless: fixes w :: "int"   and z :: "int" assumes "0 \<le> w" shows "(nat w < nat z) = (w < z)"
vs: fixes K :: "('b, 'c) ring_scheme"   and W :: "'d set"   and V :: "('b, 'd, 'e) module_scheme" assumes "subspace K W V" shows "vectorspace K V"
assms: shows "0 < Im ((b - a) * cnj (b - z))"
L_eq: fixes i :: "nat" assumes "i \<in> I" shows "L i = i"
mahler_measure_poly_def: fixes p :: "complex poly" shows "mahler_measure_poly p = cmod (lead_coeff p) * (\<Prod>a\<leftarrow>complex_roots_complex p. max 1 (cmod a))"
karatsuba_mult_poly_def: fixes f :: "'b poly"   and g :: "'b poly" shows "karatsuba_mult_poly f g = (let ff = coeffs f; gg = coeffs g; n = length ff; m = length gg in if n \<le> karatsuba_lower_bound \<or> m \<le> karatsuba_lower_bound then if n \<le> m then foldr (\<lambda>a p. smult a g + pCons (0::'b) p) ff 0 else foldr (\<lambda>a p. smult a f + pCons (0::'b) p) gg 0 else if n \<le> m then karatsuba_main gg m ff n else karatsuba_main ff n gg m)"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "c x y"   and "c y z" shows "c x z"
conj_galois_2: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "(g x \<le> y) = (x \<le> - f (- y))"
accept_Language: fixes s :: "'s" assumes "wellformed s" shows "accept s = \<oo> (Language s)"
igSwapIGOpSTR_imp_igSwapIGOp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igSwapIGOpSTR MOD" shows "igSwapIGOp MOD"
closed_segment_eq_real_ivl: fixes a :: "real"   and b :: "real" shows "closed_segment a b = (if a \<le> b then {a..b} else {b..a})"
fmdrop_fset_null: fixes m :: "('c, 'd) fmap" shows "fmdrop_fset {||} m = m"
ncoplanar_perm_13: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Coplanar C A D B"
bij_vimage_eq_inv_image: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set" assumes "bij f" shows "f -` A = inv f ` A"
pref_hd_eq': fixes p :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "p \<le>p u"   and "p \<le>p v"   and "p \<noteq> \<epsilon>" shows "hd u = hd v"
agree_Nil: fixes s :: "'d \<Rightarrow> bool" shows "agree s Mapping.empty"
delete_node_def: fixes v :: "'a"   and G :: "('a, 'b) graph_scheme" shows "delete_node v G \<equiv> \<lparr>nodes = nodes G - {v}, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> e1 \<noteq> v \<and> e2 \<noteq> v}\<rparr>"
distinct_7: fixes x51 :: "'b ltln"   and x52 :: "'b ltln" shows "true\<^sub>n \<noteq> x51 and\<^sub>n x52"
clauseOrderIrrelevant: fixes F1 :: "Literal list list"   and F :: "Literal list list"   and F' :: "Literal list list"   and F2 :: "Literal list list" shows "equivalentFormulae (F1 @ F @ F' @ F2) (F1 @ F' @ F @ F2)"
evolve: fixes ts :: "traffic"   and ts' :: "traffic"   and ts'' :: "traffic" assumes "ts \<^bold>\<Rightarrow> ts'"   and "ts' \<^bold>\<leadsto> ts''" shows "ts \<^bold>\<Rightarrow> ts''"
apply_cblinfun_code: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and S :: "complex vec list" shows "A *\<^sub>S SPAN S = (let d = length canonical_basis in SPAN (map ((*\<^sub>v) (mat_of_cblinfun A)) (filter (\<lambda>v. dim_vec v = d) S)))"
ev_at_Suc: fixes P :: "'a stream \<Rightarrow> bool"   and n :: "nat"   and \<omega> :: "'a stream" shows "ev_at P (Suc n) \<omega> = (\<not> P \<omega> \<and> ev_at P n (stl \<omega>))"
sum_invertion: fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "finite A"   and "finite B" shows "(\<Sum>x\<in>A. sum (f x) B) = (\<Sum>y\<in>B. \<Sum>x\<in>A. f x y)"
eq_trans: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<chi> :: "'b ltln" assumes "ltl_equivalence eq"   and "eq \<phi> \<psi>"   and "eq \<psi> \<chi>" shows "eq \<phi> \<chi>"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and poly :: "'b poly" assumes "\<And>x. P (x)\<^sub>p"   and "\<And>x. P (Bound x)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Add x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Sub x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Mul x1a x2a)"   and "\<And>x. P x \<Longrightarrow> P (Neg x)"   and "\<And>x1a x2a. P x1a \<Longrightarrow> P (Pw x1a x2a)"   and "\<And>x1a x2a x3. \<lbrakk>P x1a; P x3\<rbrakk> \<Longrightarrow> P (CN x1a x2a x3)" shows "P poly"
max_next_pcs_def: fixes ins :: "instr list" shows "max_next_pcs ins = {nat (int (length ins + 1) + i) |i. i \<in> offsets_is ins}"
supp_fun_app_eqvt: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "eqvt f" shows "supp (f x) \<subseteq> supp x"
DisP: fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and q :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(p \<^bold>\<or> q) at a in' set ((ps, a) # branch)"   and "new p a ((ps, a) # branch)"   and "new q a ((ps, a) # branch)"   and "A, Suc n \<turnstile> (p # ps, a) # branch"   and "A, Suc n \<turnstile> (q # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
nth_merges: fixes p :: "nat"   and ss :: "nat list list"   and ps :: "(nat \<times> nat list) list" assumes "p < length ss"   and "ss \<in> list n A"   and "\<forall>(p, t)\<in>set ps. p < n \<and> t \<in> A" shows "merges f ps ss ! p = map snd (filter (\<lambda>(p', t'). p' = p) ps) \<Squnion>\<^bsub>f\<^esub> ss ! p"
intros: fixes CAs :: "'a literal multiset list"   and n :: "nat"   and Cs :: "'a literal multiset list"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and D :: "'a literal multiset" assumes "length CAs = n"   and "length Cs = n"   and "length AAs = n"   and "length As = n"   and "n \<noteq> 0"   and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)"   and "\<forall>i<n. AAs ! i \<noteq> {#}"   and "Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs))"   and "eligible S \<sigma> As (D + negs (mset As))"   and "\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)"   and "\<forall>i<n. S (CAs ! i) = {#}" shows "ord_resolve S CAs (D + negs (mset As)) AAs As \<sigma> ((\<Sum>\<^sub># (mset Cs) + D) \<cdot> \<sigma>)"
tSup_TCons: fixes A :: "('c, 'b) tllist set"   and x :: "'c" assumes "A \<noteq> {}" shows "local.tSup (TCons x ` A) = TCons x (local.tSup A)"
sa_tb_d: shows "Q $$ (0, 0) * a + Q $$ (1, 0) * b = (Matrix.mat 1 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then a else b) * Q) $$ (0, 0)"
ord_strict_p_punit_def: fixes cmp_term :: "'a nat_term_order" shows "ord_strict_p_punit cmp_term \<equiv> ordered_term.ord_strict_p (lt_of_nat_term_order cmp_term)"
assms: fixes x :: "'a"   and y :: "'a"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "x \<succeq>[Pr] y"   and "\<alpha> + \<beta> = 1"   and "0 < \<alpha>"   and "0 < \<beta>" shows "\<alpha> *\<^sub>R x + \<beta> *\<^sub>R y \<succeq>[Pr] y"
singletonD: fixes b :: "'e"   and a :: "'e" assumes "b \<in> {a}" shows "b = a"
ord_le_eq_trans: fixes a :: "'b zmultiset"   and b :: "'b zmultiset"   and c :: "'b zmultiset" assumes "a \<subseteq>#\<^sub>z b"   and "b = c" shows "a \<subseteq>#\<^sub>z c"
fresh_6: fixes a :: "name"   and x1 :: "name"   and x2 :: "('a, 'b, 'c) psi" shows "a \<sharp> \<lparr>\<nu>x1\<rparr>x2 = a \<sharp> [x1].x2"
ubx_sp_axioms: shows "ubx_sp F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>"
mkChannels_gState_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and cs :: "channel list"   and p :: "pState" assumes "gState_inv prog g"   and "set cs \<subseteq> Collect channel_inv" shows "gState_inv prog (fst (mkChannels g p cs))"
polysub_def: fixes p :: "'a poly"   and q :: "'a poly" shows "p -\<^sub>p q = p +\<^sub>p ~\<^sub>p q"
theFT_def: fixes cl :: "('g, 'h, 'i) cmd list" shows "theFT cl \<equiv> {n. n < length cl \<and> finished (cl ! n)}"
uni_g'_2: shows "univalent ((let V_new = sorted_list_of_set (vertices (snd R) - vertices (fst R)) in set (zip V_new [n..<n + length V_new]))\<inverse>)"
less_eq_def: fixes xs :: "nat list"   and ys :: "nat list" shows "(xs \<le>\<^sub>v ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i \<le> ys ! i))"
assms: fixes i :: "'e" assumes "i \<in> CBasis" shows "continuous_on s (\<lambda>x. cblinfun_apply (f x) i)"
mg: shows "(\<lambda>x. - g x) absolutely_integrable_on {- pi..pi}"
properL_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'e) cmd list" assumes "PL cval" shows "properL cl \<equiv> cl \<noteq> [] \<and> (\<forall>c\<in>set cl. proper c)"
mat_1_fun: fixes a :: "'c"   and b :: "'c" shows "mat (1::'b) $ a $ b = (if a = b then 1::'b else (0::'b))"
valid_decomp_map_times: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and s :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "valid_decomp X ps"   and "s \<in> P[X]"   and "s \<noteq> 0" shows "valid_decomp X (map (apfst ((*) s)) ps)"
norm_point_bound_ell2: fixes x :: "'c ell2"   and i :: "'c" shows "cmod (Rep_ell2 x i) \<le> norm x"
simps_3: fixes v :: "val" shows "sub_RI (Val v) = False"
s2: shows "simple_ruleset (upper_closure (packet_assume_new (rs::'i common_primitive rule list)))"
simps: fixes a1 :: "'b set"   and a2 :: "'b \<Rightarrow> 'b" shows "cycle_decomp a1 a2 = (a1 = {} \<and> a2 = id \<or> (\<exists>I p cs. a1 = set cs \<union> I \<and> a2 = cycle_of_list cs \<circ> p \<and> cycle_decomp I p \<and> cycle cs \<and> set cs \<inter> I = {}))"
composites_are_arrows_2: shows "hseq (f \<star> g) f"
ub2: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y"
map_2: fixes f :: "'c \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c list" shows "map f (x21 # x22) = f x21 # map f x22"
mirror_\<delta>: fixes p :: "fm" assumes "iszlfm p" shows "\<delta> (mirror p) = \<delta> p"
env_delete_fun_upd: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd"   and v :: "'d" shows "env_delete x (m(x := v)) = env_delete x m"
monom_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R" shows "monom (UP R) a n \<in> carrier (UP R)"
maxim_isMaxim_support: fixes F :: "('b \<Rightarrow> 'a) set" assumes "F \<subseteq> FINFUNC"   and "const \<notin> F" shows "\<forall>f\<in>F. s.isMaxim (SUPP f) (s.maxim (SUPP f))"
unit_imp_dvd: fixes b :: "'a"   and a :: "'a" assumes "is_unit b" shows "b dvd a"
noSc: shows "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Inum (x # bs) e / real_of_int c"
map_the_restrict: fixes i :: "nat"   and A :: "nat set"   and v :: "'c option list" assumes "i \<in> A" shows "map the (restrict A v) ! i = map the v ! i"
polynomial_incl: fixes K :: "'a set"   and p :: "'a list" assumes "polynomial\<^bsub>L\<lparr>carrier := A\<rparr>\<^esub> K p" shows "set p \<subseteq> K"
cancels_to_1_at_def: fixes i :: "nat"   and l1 :: "(bool \<times> 'd) list"   and l2 :: "(bool \<times> 'd) list" shows "cancels_to_1_at i l1 l2 = (0 \<le> i \<and> 1 + i < length l1 \<and> canceling (l1 ! i) (l1 ! (1 + i)) \<and> l2 = cancel_at i l1)"
neg_2: shows "negligible {x \<in> T - S. f x \<noteq> (0::'b)}"
insert_iff: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" shows "(a \<in> Set.insert b A) = (a = b \<or> a \<in> A)"
finite_imp_foldSetD: fixes A :: "'e set"   and e :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "finite A"   and "e \<in> D"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D" shows "\<exists>x. (A, x) \<in> foldSetD D f e"
cInf_mono: fixes B :: "'a set"   and A :: "'a set"   and c :: "'a" assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"   and "\<And>a. a \<in> A \<Longrightarrow> c \<le> a"   and "B \<noteq> {}" shows "Inf A \<le> Inf B"
left_commute: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "inf b (inf a c) = inf a (inf b c)"
assms_2: fixes res :: "('a, 'b) resource"   and a :: "'a" assumes "X res"   and "a \<in> outs_\<I> \<I>" shows "lossless_spmf (run_resource res a) \<and> (\<forall>(b, res')\<in>set_spmf (run_resource res a). X res' \<or> lossless_resource \<I> res')"