hcju/isabelleps huggingface dataset:
lem8: shows "\<forall>D\<in>\<G>_F C. D \<in> Red_F_G (\<G>_Fset N) \<or> (\<exists>E\<in>N - local.Red_F_\<G> N. Prec_F_g D E C \<and> D \<in> \<G>_F E)"
nhds_enat: fixes x :: "enat" shows "nhds x = (if x = \<infinity> then INF i. principal {enat i..} else principal {x})"
least_contains': fixes A :: "'d set" assumes "A \<noteq> {}" shows "least A \<in> A"
nprv_clear3_1: fixes \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>1 :: "'fmla" assumes "nprv {\<phi>2, \<phi>3} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3} \<psi>"
rec_minus_def: shows "rec_minus = rec_swap (PR (recf.Id 1 0) (CN rec_predecessor [recf.Id 3 1]))"
MVT0_within: fixes t :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real \<Rightarrow> real"   and s :: "real" assumes "\<And>x. x \<in> {0..t} \<Longrightarrow> (f has_derivative f' x) (at x within {0..t})"   and "\<And>x. x \<in> {0..t} \<Longrightarrow> 0 \<le> f' x s"   and "0 < s \<and> s \<le> t"   and "0 < t" shows "f 0 \<le> f s"
ordLeq_ordIso_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r \<le>o r'"   and "r' =o r''" shows "r \<le>o r''"
refl: fixes x :: "'b" assumes "x \<in> alts" shows "Pareto R x x"
winding_number_pos_lt_lemma: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and e :: "real" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "0 < e"   and "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" shows "0 < Re (winding_number \<gamma> z)"
rat_number_expand_3: fixes k :: "num" shows "numeral k = Fract (numeral k) 1"
f: shows "f = (\<lambda>s. nat (s ''a'') + 1)"
sort_eqvI: fixes cs :: "(String.literal \<times> String.literal) set"   and s1 :: "String.literal set"   and s2 :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_leq cs s1 s2"   and "sort_leq cs s2 s1" shows "sort_eqv cs s1 s2"
AE: shows "AE \<omega> in T c. sfirst X (c ## \<omega>) < \<infinity>"
height_mkt_bal_l: fixes l :: "'b tree"   and r :: "'b tree"   and n :: "'b" assumes "height l = height r + 2"   and "avl l"   and "avl r" shows "height (mkt_bal_l n l r) = height r + 2 \<or> height (mkt_bal_l n l r) = height r + 3"
distinct_27: fixes x3 :: "('a, 'b, 'c) formula"   and x61 :: "('a, 'b, 'c) hp"   and x62 :: "('a, 'b, 'c) formula" shows "! x3 \<noteq> (\<langle> x61 \<rangle> x62)"
has_integral_unique: fixes f :: "'c \<Rightarrow> 'd"   and k1 :: "'d"   and i :: "'c set"   and k2 :: "'d" assumes "(f has_integral k1) i"   and "(f has_integral k2) i" shows "k1 = k2"
tangent_space_def: fixes p :: "'a" shows "tangent_space p = {X. is_derivation X p \<and> extensional0 diff_fun_space X}"
bounded_iff: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" shows "(ordered_powerprod_lin.max b c \<preceq> a) = (b \<preceq> a \<and> c \<preceq> a)"
cod_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "A_B.cod f = (if A_B.arr f then A_B.MkIde (A_B.Cod f) else A_B.null)"
g: assumes "n \<noteq> 0" shows "eval g [f \<triangleright> n] \<down>= (if the (inconsist (amalg01 f) (f \<triangleright> n)) = e_length (f \<triangleright> n) then amalg01 f else the (eval (Cn 1 r_ifz [p, at1, at0]) [f \<triangleright> n]))"
ide_in_hom_2: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "\<guillemotleft>f : f \<Rightarrow>\<^sub>S f\<guillemotright>"
inv_upto_Suc: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat" assumes "local.inv_upto p A m"   and "\<And>i. i < n \<Longrightarrow> p A i m" shows "local.inv_upto p A (Suc m)"
T_def: shows "T \<equiv> n * (n - 1) div 2"
list_all2_Nil_iff: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" assumes "list_all2 R xs ys" shows "(xs = []) = (ys = [])"
monic_prod_list: fixes as :: "'a poly list" assumes "\<And>a. a \<in> set as \<Longrightarrow> monic a" shows "monic (prod_list as)"
li_le_dim_2: fixes A :: "'c set" assumes "fin_dim"   and "A \<subseteq> carrier V"   and "lin_indpt A" shows "card A \<le> local.dim"
distinct_33: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> Await x71 x72"
terminates_dynCall_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>dynCall init p return c \<down> Normal s" obtains "\<Gamma>\<turnstile>call init (p s) return c \<down> Normal s"
res_rc3: fixes y :: "'a"   and x :: "'a" shows "y \<rightarrow> y \<cdot> (y \<rightarrow> y \<cdot> x) = y \<rightarrow> y \<cdot> x"
f: fixes x :: "'a" assumes "x \<in> space N" shows "domain (f x) \<in> I \<and> (\<forall>i\<in>domain (f x). (f x)\<^sub>F i \<in> space (M i))"
inj_vfsequence_of_vlist: shows "inj vfsequence_of_vlist"
word_and_or_mask_aligned: fixes a :: "'b word"   and n :: "nat"   and b :: "'b word" assumes "is_aligned a n"   and "b \<le> mask n" shows "a + b = a OR b"
subsetCE: fixes A :: "'d set"   and B :: "'d set"   and c :: "'d" assumes "A \<subseteq> B" obtains "\<not> c \<in> A"   | "c \<in> B"
assms: fixes \<alpha> :: "act"   and Q' :: "ccs" assumes "Q \<longmapsto>\<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
is_K2_isometry_hyp2_S: fixes p :: "proj2"   and J :: "cltn2" assumes "p \<in> hyp2 \<union> S"   and "is_K2_isometry J" shows "apply_cltn2 p J \<in> hyp2 \<union> S"
splitting_sublemma_gen: fixes C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and H :: "'b set"   and K :: "'b set" assumes "exact_seq ([C, B, A], [g, f])"   and "f ` carrier A = H"   and "subgroup K B"   and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}"   and "H <#>\<^bsub>B\<^esub> K = carrier B" shows "g \<in> Group.iso (subgroup_generated B K) (subgroup_generated C (g ` carrier B))"
select_convs_3: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cTimeF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cTimeF"
red_def: shows "red \<equiv> \<lambda>P. {((xb, xa, x), xe, xd, xc). redp P xb xa x xe xd xc}"
norm_triangle_lt: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y < e" shows "norm (plus x y) < e"
inverse_arrows_def: fixes f :: "'a"   and g :: "'a" shows "C.inverse_arrows f g \<equiv> C.ide (g \<cdot> f) \<and> C.ide (f \<cdot> g)"
parallel_pair_diagram_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b" shows "parallel_pair_diagram_axioms C f0 f1 \<equiv> partial_magma.arr C f0 \<and> partial_magma.arr C f1 \<and> partial_magma.dom C f0 = partial_magma.dom C f1 \<and> partial_magma.cod C f0 = partial_magma.cod C f1"
tmps_distinct: shows "tmps_distinct ts\<^sub>s\<^sub>b"
nres_rel_def: fixes R :: "('g \<times> 'h) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
keys_def: fixes t :: "('a, 'b) rbt" shows "RBT_Impl.keys t = map fst (RBT_Impl.entries t)"
sorted_sort_key: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "sorted (map f (sort_key f xs))"
col_per2__per: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and X :: "'p"   and P :: "'p" assumes "A \<noteq> B"   and "Col A B C"   and "Per A X P"   and "Per B X P" shows "Per C X P"
part_witness_pow: fixes X :: "part_witness set" shows "X \<in> set ` set (subseqs [part_witness.a, part_witness.b, part_witness.c])"
l_get_parent_wf_axioms: shows "l_get_parent_wf DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent"
not_bot: fixes A :: "'a pred" assumes "A \<noteq> \<bottom>" obtains x :: "'a" where "eval A x"
map_prod_surj_on: fixes f :: "'h \<Rightarrow> 'g"   and A :: "'h set"   and A' :: "'g set"   and g :: "'j \<Rightarrow> 'i"   and B :: "'j set"   and B' :: "'i set" assumes "f ` A = A'"   and "g ` B = B'" shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
hd_Cons_tl: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "hd xs # tl xs = xs"
f: fixes x :: "'a" assumes "x \<in> S" shows "f x = exp (g x)"
greaterThanLessThan_def: fixes l :: "'a"   and u :: "'a" shows "{l<..<u} \<equiv> {l<..} \<inter> {..<u}"
eval_ternary_idempotence_Not: fixes a :: "ternaryvalue" shows "eval_ternary_Not (eval_ternary_Not a) = a"
bi_unique_mod_ring_rel64_2: shows "left_unique mod_ring_rel64"
reduced_word_for_imp_reduced_word: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A a as" shows "monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero as) as"
simps_1: fixes A :: "'b \<times> ('c \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)" shows "steps A [] = Id"
bounded_linear_vec_nth: fixes i :: "'d" shows "bounded_linear (\<lambda>x. x $ i)"
subterms_of_instances: fixes p :: "indices list" shows "\<forall>u v u' s. u = v \<lhd> s \<longrightarrow> subterm u p u' \<longrightarrow> (\<exists>x q1 q2. is_a_variable x \<and> subterm (x \<lhd> s) q1 u' \<and> subterm v q2 x \<and> p = q2 @ q1) \<or> (\<exists>v'. \<not> is_a_variable v' \<and> subterm v p v' \<and> u' = v' \<lhd> s)"
DERIV_continuous_on: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative D x) (at x within s)" shows "continuous_on s f"
Inum_C: fixes i :: "nat"   and vs :: "nat list" shows "Inum (C i) vs = i"
ltl_Lazy_llist: fixes xs :: "unit \<Rightarrow> ('a \<times> 'a llist) option" shows "ltl (Lazy_llist xs) = Lazy_llist (\<lambda>_. case xs () of None \<Rightarrow> None | Some (x, ys) \<Rightarrow> force ys)"
path_append_elim: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a list"   and s :: "'a list"   and p :: "'b" assumes "transition_system.path execute enabled (r @ s) p" obtains "transition_system.path execute enabled r p" and "transition_system.path execute enabled s (fold execute r p)"
list_of_oalist_OAlist: fixes xs :: "('a \<times> 'b) list \<times> 'a key_order" shows "list_of_oalist (OAlist xs) = sort_oalist_ko xs"
closed_epigraph_lines: fixes S :: "('b \<times> 'c) set"   and x :: "'b" assumes "closed S" shows "closed {z. (x, z) \<in> S}"
plus_period: fixes x :: "int" shows "gauss_sum_int (x + int n) = gauss_sum_int x"
elims_1: fixes x :: "nat"   and xa :: "'b btree"   and y :: "bool" assumes "almost_order x xa = y" obtains k :: "nat" where "x = ka__" and "xa = Leaf" and "y"   | k :: "nat" and ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = k" and "xa = Node ts t" and "y = (length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t)"
partition_set_of_intersecting_sets_by_card: fixes A :: "'a set" assumes "finite A" shows "{B. A \<inter> B \<noteq> {}} = (\<Union>n\<in>{1..card A}. {B. card (A \<inter> B) = n})"
MM: shows "Ik.MModel (ProblemIkTpart_TEModel.intT prot protFw eintT eintF) (ProblemIkTpart_TEModel.intF arOf resOf prot protFw eintT eintF) (ProblemIkTpart_TEModel.intP parOf prot protFw eintT eintF eintP)"
ltrm_prod_ltrm: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "monom P ((monom P (p (deg R p)) (deg R p) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)) (deg R (monom P (p (deg R p)) (deg R p) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)))) (deg R (monom P (p (deg R p)) (deg R p) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q))) = monom P (p (deg R p)) (deg R p) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)"
le_add_same_cancel2: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus b a) = less_eq zero b"
up_def: shows "up \<equiv> hom_induced (int n) (lsphere n) (equator n) (nsphere n) (equator n) id zp"
hcomp_def: fixes g :: "'a"   and f :: "'a" shows "g \<star> f = (if S.arr f \<and> S.arr g \<and> trg\<^sub>B f = src\<^sub>B g then g \<star>\<^sub>B f else S.null)"
listsetD_1: fixes ys :: "'a list"   and xs :: "'a set list" assumes "ys \<in> listset xs" shows "length ys = length xs"
nu_par: fixes x :: "'a"   and y :: "'a" shows "\<nu> (x \<parallel> y) = d (\<tau> x) \<cdot> \<nu> y + d (\<tau> y) \<cdot> \<nu> x + \<nu> x \<parallel> \<nu> y"
berlekamp_resulting_mat_closed_3: fixes u :: "'b mod_ring poly" shows "dim_col (berlekamp_resulting_mat u) = degree u"
lm078: shows "injectionsUniverse \<subseteq> runiqs"
negligible_insert: fixes a :: "'a"   and s :: "'a set" shows "negligible (insert a s) = negligible s"
ldvd: fixes x :: "int" shows "\<Lambda> dvd x * (\<Lambda> * (\<v> - 1) div (\<k> - 1) * (\<k> - 1) + \<Lambda>)"
eq: shows "ko.lt (key_order_of_nat_term_order_inv cmp_term) = ord_term_strict_conv"
Cons_2: shows "set (a # as) \<subseteq> set bs"
fsubst_induct: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and P :: "('a \<Rightarrow> ('b, 'a) Term.term) \<Rightarrow> bool" assumes "finite (subst_domain \<delta>)"   and "P Var"   and "\<And>\<theta> v t. \<lbrakk>finite (subst_domain \<theta>); v \<notin> subst_domain \<theta>; t \<noteq> Var v; P \<theta>\<rbrakk> \<Longrightarrow> P (\<theta>(v := t))" shows "P \<delta>"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (- x) = - lookup x"
hyps: fixes s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option" assumes "\<Gamma>,High \<turnstile> c'"   and "\<langle>c',s1\<rangle> \<rightarrow>* \<langle>Skip,s2\<rangle>" shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"
6: shows "w\<^sup>T\<^sup>+ \<le> - w"
Try_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and C :: "String.literal"   and V :: "nat" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (try e catch(C V) e') h (stk, loc, Suc (Suc (length (compE2 e) + pc)), xcp) (stk', loc', Suc (Suc (length (compE2 e) + pc')), xcp')"
ring_def: fixes R :: "('c, 'd) ring_scheme" shows "ring R \<equiv> abelian_group R \<and> Group.monoid R \<and> ring_axioms R"
moebius_circline_id: fixes H :: "circline" shows "moebius_circline id_moebius H = H"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and \<mu> :: "'b" assumes "arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>"   and "identity_arrow_of_tabulations_in_maps_axioms V \<mu>" shows "identity_arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>"
simps_1: fixes S :: "Runs.var set"   and x :: "Runs.var set" shows "option_inter S (Some x) = Some (x \<inter> S)"
le_trans: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "x \<le> y"   and "y \<le> z"   and "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "x \<le> z"
naturality: fixes f :: "'a" assumes "B.arr f" shows "B.\<ll> (B.cod f) \<cdot>\<^sub>B B.L f = B.map f \<cdot>\<^sub>B B.\<ll> (B.dom f)"
hyps_3: shows "\<Gamma> \<turnstile> S <: T"
mult_when: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and P :: "bool" assumes "class.mult_zero times zero" shows "times a (zero.when zero b P) = zero.when zero (times a b) P"
goal3_4: shows "Rmi_g (Suc (Suc n)) (IF v t e) bdd"
enumerate_map_upt: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and m :: "nat" shows "List.enumerate n (map f [n..<m]) = map (\<lambda>k. (k, f k)) [n..<m]"
r_smn_def: fixes n :: "nat"   and m :: "nat" shows "r_smn n m \<equiv> Cn (Suc m) r_prod_encode [r_constn m 3, Cn (Suc m) r_prod_encode [r_constn m n, Cn (Suc m) r_prod_encode [r_constn m (encode (r_universal (n + m))), r_smn_aux n m]]]"
coeff_eq_0: fixes p :: "'b poly"   and n :: "nat" assumes "degree p < n" shows "coeff p n = (0::'b)"
replicate_Suc: fixes n :: "nat"   and x :: "'a" shows "x \<up> Suc n = x # x \<up> n"
obsne: shows "obs \<sigma> \<noteq> obs \<sigma>'"
comp_converter_id_left: fixes conv :: "('a, 'b, 'c, 'd) converter" shows "comp_converter id_converter conv = conv"
poly_div_add_left: fixes x :: "'b poly"   and y :: "'b poly"   and z :: "'b poly" shows "(x + y) div z = x div z + y div z"
normalize_lcm: fixes a :: "'a"   and b :: "'a" shows "normalize (lcm a b) = lcm a b"
1: shows "(\<lambda>x. x /\<^sub>R norm x) ` B \<subseteq> S"
emp_shift_rev: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and I :: "nat set" assumes "PL_Indis cval indis" shows "({} = PL_Indis.shift cl n I) = (I = {})"
N: fixes m :: "nat"   and n :: "nat" assumes "N \<le> m"   and "N \<le> n" shows "norm (sum f {..<m} - sum f {..<n}) < e"
a_rcos_const: fixes h :: "'a" assumes "h \<in> H" shows "H +> h = H"
clinear: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "clinear f"
ih: shows "cnf (foldr (\<^bold>\<and>) fs (\<^bold>\<not> \<bottom>)) = \<Union> (cnf ` set fs)"
sigma_finite_measure_axioms: fixes M :: "'a pmf" shows "sigma_finite_measure (measure_pmf M)"
smc_0_is_arr_iff: fixes \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<^bsub>smc_0\<^esub> \<BB> = False"
enat_of_option_inj: fixes x :: "nat option"   and y :: "nat option" shows "(Dijkstra_Abstract.enat_of_option x = Dijkstra_Abstract.enat_of_option y) = (x = y)"
ESem_considers_fv': fixes e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho> f|` fv e\<^esub>"
simps_1: fixes k :: "'a"   and Ks :: "'a list"   and u :: "'a"   and Us :: "'a list" shows "combine (k # Ks) (u # Us) = k \<otimes> u \<oplus> combine Ks Us"
rt_fresher_addpreRT: fixes ip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and npre :: "nat set" assumes "ip \<in> kD rt1" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> the (addpreRT rt1 ip npre)"
reduce_root_order: fixes k :: "nat"   and t :: "'b btree" assumes "0 < k"   and "almost_order k t" shows "root_order k (reduce_root t)"
gcd_fract_def: shows "gcd \<equiv> normalization_euclidean_semiring_class.gcd"
Nonce_synth: fixes n :: "nat"   and H :: "msg set" assumes "Nonce n \<in> synth H" obtains "Nonce n \<in> H"
Quotient_rel_abs: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and r :: "'e"   and s :: "'e" assumes "Quotient R Abs Rep T"   and "R r s" shows "Abs r = Abs s"
circ_plus_sub: fixes x :: "'a" shows "x\<^sup>\<circ> * x \<le> x * x\<^sup>\<circ>"
map_poly_poly_square_subst: fixes f :: "'a poly" shows "map_poly hom (poly_square_subst f) = poly_square_subst (map_poly hom f)"
perm_simps_3: fixes p :: "perm"   and fmla :: "fmla" shows "p \<bullet> neg fmla = neg (p \<bullet> fmla)"
has_field_derivative_subset: fixes f :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "(f has_field_derivative y) (at x within s)"   and "t \<subseteq> s" shows "(f has_field_derivative y) (at x within t)"
assoc'_closed: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "Arr f \<and> B.ide f"   and "Arr g \<and> B.ide g"   and "Arr h \<and> B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "Arr (B.inv \<a>\<^sub>B[f, g, h])"
simps_1: fixes f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "pred \<Up>f (cf # cfs) = True"
mgp_edges_disj: fixes g :: "graph"   and f :: "face"   and f' :: "face"   and uv :: "nat \<times> nat" assumes "minGraphProps g"   and "f \<noteq> f'"   and "f \<in> \<F> g"   and "f' \<in> \<F> g"   and "uv \<in> \<E> f" shows "uv \<notin> \<E> f'"
mni1_def: shows "mni1 = m + n - Suc i"
acc_run_run: fixes r :: "nat \<Rightarrow> 'v" assumes "G'.is_acc_run r" shows "G'.is_run r"
reachable_Assign: fixes x :: "char list"   and a :: "aexp" shows "reachable (x ::= a) = {x ::= a, SKIP}"
cocomputation_def: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration" shows "cocomputation channel trans send recv init \<equiv> distributed_system channel \<and> cocomputation_axioms channel trans send recv init"
smtoms_eq_hlp: fixes r :: "(32, 'b) simple_match_scheme"   and a :: "char list option"   and b :: "protocol"   and c :: "(16 word \<times> 16 word) option"   and d :: "(16 word \<times> 16 word) option"   and f :: "char list option"   and g :: "protocol"   and h :: "(16 word \<times> 16 word) option"   and i :: "(16 word \<times> 16 word) option" shows "(simple_match_to_of_match_single r a b c d = simple_match_to_of_match_single r f g h i) = (a = f \<and> b = g \<and> c = h \<and> d = i)"
iso_iff_section_and_retraction: fixes f :: "'c" shows "C.iso f = (C.section f \<and> C.retraction f)"
assms_4: shows "(i, Expunge e1 mo i) \<in> set (node_deliver_messages xs)"
Derivation_empty_no_terminals: fixes u :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and t :: "symbol" assumes "Derivation u D []"   and "t \<in> set u" shows "is_nonterminal t"
setdiff_irrelevant: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A" shows "sum g (A - {x. g x = (0::'b)}) = sum g A"