hcju/isabelleps huggingface dataset:
fls_times_nth_1: fixes f :: "'b fls"   and g :: "'b fls"   and n :: "int" shows "fls_nth (f * g) n = (\<Sum>i = fls_subdegree f + fls_subdegree g..n. fls_nth f (i - fls_subdegree g) * fls_nth g (fls_subdegree g + n - i))"
count_mset_set_1: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<in> A" shows "count (mset_set A) x = 1"
1: shows "[\<lparr>O!,x\<rparr> \<^bold>& \<lparr>O!,y\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>) in v]"
igAbsCongS_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igAbsCongS MOD \<equiv> \<forall>xs x x' y s X X'. isInBar (xs, s) \<and> igWls MOD s X \<and> igWls MOD s X' \<longrightarrow> igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSwap MOD xs y x X = igSwap MOD xs y x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
sup_comm: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
ends_in_segment_1: fixes a :: "'a"   and b :: "'a" shows "a \<in> closed_segment a b"
wls_swap_ident: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "wls s X" shows "X #[x \<and> x]_xs = X"
coprime_cong_mult: fixes a :: "'b"   and b :: "'b"   and m :: "'b"   and n :: "'b" assumes "[a = b] (mod m)"   and "[a = b] (mod n)"   and "coprime m n" shows "[a = b] (mod m * n)"
sig_red_single_sing_regI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "sig_red_single (\<preceq>\<^sub>t) top_tail p q f t"
a_def: shows "a \<equiv> \<Sum>x\<in>A. prod f (A - {x})"
infliveness_def: fixes A :: "'b set"   and P :: "'b llist set" shows "infliveness A P \<equiv> \<forall>t\<in>A\<^sup>\<star>. \<exists>s\<in>A\<^sup>\<omega>. t @@ s \<in> P"
finished_inf: fixes step :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and final :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "semantics step final"   and "Inf.inf step s" shows "\<not> finished step s"
lv_rel_nth: fixes xs :: "real list"   and x :: "'b"   and b :: "'b" assumes "(xs, x) \<in> lv_rel"   and "b \<in> Basis" shows "xs ! index Basis_list b = x \<bullet> b"
extended_Gromov_product_boundary_inside: fixes e :: "'a"   and y :: "'a Gromov_completion"   and x :: "'a" shows "extended_Gromov_product_at e y (to_Gromov_completion x) = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e (v n) x)) |v. (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
finite_dimensional_vector_space_axioms: shows "finite_dimensional_vector_space (*s) Basis"
inter_rel_def: fixes AA :: "('d \<times> 'f set) set"   and BB :: "('e \<times> 'f set) set" shows "\<langle>AA, BB\<rangle>inter_rel = {((a, b), A \<inter> B) |a b A B. (a, A) \<in> AA \<and> (b, B) \<in> BB}"
assms_5: shows "fmrestrict_set vsb fm = fmrestrict_set (vsb \<inter> vs) fmb"
invertible_Units: fixes A :: "'a mat"   and b :: "'b" assumes "invertible_mat A" shows "A \<in> Units (ring_mat TYPE('a) (dim_row A) b)"
extreme_point_of_Int_supporting_hyperplane_le: fixes S :: "'b set"   and a :: "'b"   and b :: "real"   and c :: "'b" assumes "S \<inter> {x. a \<bullet> x = b} = {c}"   and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b" shows "c extreme_point_of S"
abc: shows "reduce_system p ([1], base_case_info) = (a, b, c)"
nsum_memrTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>i\<le>n. f i \<in> carrier R) \<longrightarrow> (\<forall>l\<le>n. \<Sigma>\<^sub>e R f l \<in> carrier R)"
simps: fixes \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "subst_range \<sigma> = \<sigma> ` subst_domain \<sigma>"
RDirProd_list_zero: fixes Rs :: "('c, 'd) ring_scheme list" shows "\<zero>\<^bsub>RDirProd_list Rs\<^esub> = foldr (\<lambda>R. (#) \<zero>\<^bsub>R\<^esub>) Rs []"
S3_live: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> WF(RPCNext crCh rmCh rst p)_r p \<longrightarrow> (S3 rmhist p \<leadsto> S4 rmhist p \<or> S6 rmhist p)"
rep_eq: fixes x :: "'c fset"   and xa :: "'c fset" shows "(x |\<subseteq>| xa) = (fset x \<subseteq> fset xa)"
r1_2: shows "R1 = stabilizer G \<phi> x #> r1"
gbinomial_nneg: fixes a :: "int"   and k :: "nat" assumes "0 \<le> a" shows "a gchoose k = int (nat a choose k)"
Sup_specs_Sup_spec: fixes d :: "nat" assumes "d = DIM('c)" shows "(Sup_specs d, Sup_spec) \<in> \<langle>lv_rel\<rangle>set_rel \<rightarrow> \<langle>lv_rel\<rangle>nres_rel"
adj_dual: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "f \<stileturn> g = \<partial>\<^sub>F g \<stileturn> \<partial>\<^sub>F f"
assms_4: shows "(i, Append i e1) \<in> set (node_deliver_messages xs)"
mem_existence_ivl_iv_defined_1: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "t0 \<in> T"
always_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<box>P = (\<lambda>\<sigma>. \<forall>i. P (\<sigma> |\<^sub>s i))"
strict_prefix_eq_exists: fixes s :: "'b list"   and t :: "'b list" shows "strict_prefix s t = (\<exists>xs. s @ xs = t \<and> 0 < length xs)"
simps_1: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree" shows "BinomialHeapStruc.getMinTree [t] = t"
I'2: fixes t :: "real" assumes "0 < t"   and "t \<notin> D" shows "(I has_vector_derivative f t) (at t)"
zpos_apos: fixes n :: "int" shows "(0 \<le> n) = (0 \<le> ant n)"
intros_6: fixes u :: "'d \<Rightarrow> 'c"   and r :: "'d"   and c :: "'d"   and d :: "'c" assumes "u r - u c < d" shows "dbm_entry_val u (Some r) (Some c) (Lt d)"
cond_prob_def: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
wellformed_inc_dot: fixes x :: "item"   and d :: "nat" assumes "wellformed_item x"   and "item_dot x + d \<le> length (item_rhs x)" shows "wellformed_item (inc_dot d x)"
prev: shows "if bounded S then connected (frontier S) else \<forall>C\<in>components (frontier S). \<not> bounded C"
WtNFT_1_WtFT: fixes cl :: "('d, 'e, 'f) cmd list" assumes "0 < length cl" shows "WtNFT cl = 1 - WtFT cl"
integrable_on_subcbox: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "f integrable_on S"   and "cbox a b \<subseteq> S" shows "f integrable_on cbox a b"
c'_def: shows "c' = (if conv_abscissa f = - \<infinity> then c - 2 else (c - 1 + real_of_ereal (conv_abscissa f)) / 2)"
measurable_Pair2_compose: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and M :: "'c measure"   and M1 :: "'d measure"   and M2 :: "'e measure"   and h :: "'f \<Rightarrow> 'c"   and N :: "'f measure" assumes "(\<lambda>x. (f x, g x)) \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"   and "h \<in> N \<rightarrow>\<^sub>M M" shows "(\<lambda>x. g (h x)) \<in> N \<rightarrow>\<^sub>M M2"
simps: fixes x :: "'b"   and xs :: "'b list" shows "List.last (x # xs) = (if xs = [] then x else List.last xs)"
card_B: fixes m :: "nat"   and f :: "nat \<Rightarrow> vote" shows "card {x \<in> {1..m}. f x = B} = m - card {x \<in> {1..m}. f x = A}"
exec_Normal_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Basic f,Normal s\<rangle> \<Rightarrow> t" obtains "t = Normal (f s)"
nat_term_compD4': fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and a :: "'a"   and v :: "'a"   and b :: "'a" assumes "comparator cmp"   and "nat_term_comp cmp"   and "fst (rep_nat_term u) = fst (rep_nat_term a)"   and "fst (rep_nat_term v) = fst (rep_nat_term b)"   and "snd (rep_nat_term u) = snd (rep_nat_term v)"   and "snd (rep_nat_term a) = snd (rep_nat_term b)" shows "cmp u v = cmp a b"
vc_mono: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and C :: "acom" assumes "\<forall>s. P s \<longrightarrow> P' s"   and "vc C P" shows "vc C P'"
Cod_Inv: fixes t :: "'c local.term" assumes "Can t" shows "Cod (Inv t) = Dom t"
m_def: shows "m = (if space M = {} then 0 else MAX x\<in>space M. norm (s i x))"
simps_1: fixes f :: "'fsym" shows "GE_resOf (Oldf f) = resOf f"
update_rule: fixes p :: "'b ref"   and y :: "'b"   and x :: "'b" shows "<p \<mapsto>\<^sub>r y> p := x <\<lambda>r. p \<mapsto>\<^sub>r x>"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "zero_hom hom"   and "semigroup_add_hom hom" shows "monoid_add_hom hom"
homeomorphic_maps_compose: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and h :: "'d \<Rightarrow> 'c"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e"   and k :: "'e \<Rightarrow> 'd" assumes "homeomorphic_maps X Y f h \<and> homeomorphic_maps Y X'' g k" shows "homeomorphic_maps X X'' (g \<circ> f) (h \<circ> k)"
landau_trans_34: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
intro: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" assumes "\<And>R i Ri'. \<lbrakk>pref_profile_wf agents alts R; i \<in> agents; total_preorder_on alts Ri'\<rbrakk> \<Longrightarrow> \<not> social_decision_scheme.manipulable_profile sds R i Ri'" shows "strategyproof_sds_axioms agents alts sds"
context_typing_1: fixes \<Gamma> :: "binding list"   and u :: "trm"   and T :: "POPLmarkRecord.type"   and E :: "trm \<Rightarrow> trm"   and t :: "trm"   and t' :: "trm" assumes "\<Gamma> \<turnstile> u : T"   and "E \<in> ctxt"   and "u = E t"   and "\<And>T\<^sub>0. \<Gamma> \<turnstile> t : T\<^sub>0 \<Longrightarrow> \<Gamma> \<turnstile> t' : T\<^sub>0" shows "\<Gamma> \<turnstile> E t' : T"
conv_radius_mult_power: fixes c :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "conv_radius (\<lambda>n. c ^ n * f n) = conv_radius f / ereal (norm c)"
NF_no_step: fixes a :: "'b"   and A :: "('b \<times> 'b) set" assumes "a \<in> NF A" shows "\<forall>b. (a, b) \<notin> A"
match_rews_1: fixes k :: "'r match" shows "match_Null\<cdot>\<bottom>\<cdot>k = \<bottom>"
SourcesS3_L2: shows "Sources level2 sS3 = {sS2}"
connected_component_def: fixes S :: "'c set"   and x :: "'c"   and y :: "'c" shows "connected_component S x y \<equiv> \<exists>T. connected T \<and> T \<subseteq> S \<and> x \<in> T \<and> y \<in> T"
Un_mono: fixes A :: "'e set"   and C :: "'e set"   and B :: "'e set"   and D :: "'e set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<union> B \<subseteq> C \<union> D"
taylor_deg_1_eval': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int"   and fa :: "nat \<Rightarrow> int"   and f'a :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c = shift 2 (taylor a f) \<bullet> b"   and "fa = f \<bullet> a"   and "f'a = deriv f a" shows "f \<bullet> (a \<oplus> b) = fa \<oplus> f'a \<otimes> b \<oplus> c \<otimes> b [^] 2"
abs_square_le_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times x 2) one = less_eq (abs x) one"
p: shows "p = Polynomial.smult c (\<Prod>(a, i)\<in>set fs. a ^ Suc i)"
of_real_eq_0_iff: fixes x :: "real" shows "(of_real x = (0::'b)) = (x = 0)"
Rn: shows "card {x. x ^ n = 1} \<le> n"
get_tag_name_impl: shows "get_tag_name = CD.a_get_tag_name"
l_minus: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
fmlookup: fixes x :: "('c, 'd) fmap" shows "($$) x \<in> {m. finite (dom m)}"
is_unique: fixes I :: "'a set"   and J :: "'a set" assumes "max_lideal I A add mult zero one"   and "max_lideal J A add mult zero one" shows "I = J"
prappend_signed_list_assoc: fixes xs :: "('a \<times> bool) list"   and ys :: "('a \<times> bool) list"   and zs :: "('a \<times> bool) list" assumes "proper_signed_list xs"   and "proper_signed_list ys"   and "proper_signed_list zs" shows "prappend_signed_list (prappend_signed_list xs ys) zs = prappend_signed_list xs (prappend_signed_list ys zs)"
simps_2: fixes s :: "'b set"   and ss :: "'b set list"   and ts :: "'b set set" shows "partitioning (s # ss) ts = partitioning ss (addSubsetSet s ts)"
stable_rank_alt_def: fixes x :: "nat"   and n :: "nat"   and j :: "nat" shows "(rank x n = Some j \<and> stable_rank x j) = (\<forall>m\<ge>n. rank x m = Some j)"
invertible_fst_Gauss_Jordan_upt_k_PA: fixes A :: "(('a, 'c) vec, 'b) vec"   and k :: "nat" shows "invertible (fst (Gauss_Jordan_upt_k_PA A k))"
hom_det: fixes A :: "'a mat" shows "det (mat_hom A) = to_fract (det A)"
transitive_closure_trans_6: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
PO_l2_inv4: shows "reach l2 \<subseteq> l2_inv4"
hn_refine_cons_res: fixes \<Gamma> :: "assn"   and f :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and g :: "'b nres"   and R' :: "'b \<Rightarrow> 'a \<Rightarrow> assn" assumes "hn_refine \<Gamma> f \<Gamma>' R g"   and "\<And>a c. R a c \<Longrightarrow>\<^sub>t R' a c" shows "hn_refine \<Gamma> f \<Gamma>' R' g"
orbit_equivalent_def: fixes a :: "'a"   and b :: "'a" shows "(a \<sim>\<^sub>\<K> b) = (Orb a = Orb b)"
isfields_def: fixes P :: "'b prog"   and C :: "char list" shows "isfields P C \<equiv> filter (\<lambda>((F, D), b, T). b = Static \<and> D = C) (fields P C)"
Rep_binding_inverse: fixes x :: "binding" shows "Abs_binding (Rep_binding x) = x"
neq_PInf_trans: fixes y :: "ereal"   and x :: "ereal" assumes "y \<noteq> \<infinity>"   and "x \<le> y" shows "x \<noteq> \<infinity>"
assms: shows "S = {wd \<in> space (completion (lborel_f weight_space_dim)). \<exists>ws Z. Z < r ^ N_half \<and> (\<forall>inputs. input_sizes (deep_model_l rs) = map dim_vec inputs \<longrightarrow> evaluate_net (insert_weights shared_weights (deep_model_l rs) wd) inputs = evaluate_net (insert_weights shared_weights (shallow_model (rs ! 0) Z (last rs) (2 * N_half - 1)) ws) inputs)}"
tape_of_nl_cons_app1: fixes a :: "nat"   and xs :: "nat list"   and b :: "nat" shows "<a # xs @ [b]> = Oc \<up> Suc a @ Bk # <xs @ [b]>"
numbering_translation_for_phi: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "\<psi> \<in> \<P>\<^sup>2" shows "\<exists>c\<in>\<R>. \<forall>i. \<psi> i = \<phi> (the (c i))"
exhaust: fixes y :: "cmp_result" obtains "y = LT"   | "y = EQ"   | "y = GT"
s'_pos: fixes x :: "real" assumes "0 \<le> x"   and "x < ego2.t_stop"   and "0 \<le> v\<^sub>e"   and "a\<^sub>e \<le> 0" shows "0 < ego2.s' x"
subseq_append: fixes xs :: "'a list"   and zs :: "'a list"   and ys :: "'a list" shows "subseq (xs @ zs) (ys @ zs) = subseq xs ys"
lucas_lehmer_correct': fixes p :: "nat" shows "prime (2 ^ p - 1) = (prime p \<and> (p = 2 \<or> lucas_lehmer_test p))"
ballH_2: shows "cball s (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> S \<subseteq> cball s \<epsilon>\<^sub>2 \<inter> S"
d_IN_split_Vertex: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" shows "d_IN (split f) (Vertex x) = d_IN f x"
unit_ball_vol_even: fixes n :: "nat" shows "unit_ball_vol (real (2 * n)) = pi ^ n / fact n"
inducts: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "wave \<Gamma> f" obtains "separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f)" and "\<And>x. x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<Longrightarrow> d_OUT f x = 0"
bv_to_nat_Nil: shows "bv_to_nat [] = 0"
induct: fixes P :: "'a err \<Rightarrow> bool"   and err :: "'a err" assumes "P Err"   and "\<And>x. P (OK x)" shows "P err"
subadditiveI: fixes u :: "nat \<Rightarrow> real" assumes "\<And>m n. u (m + n) \<le> u m + u n" shows "Fekete.subadditive u"
repcs_valid: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg" shows "repcs s cfg \<in> MDP.valid_cfg"
x_as_monom: shows "[:0::'c, 1::'c:] = monom (1::'c) 1"
set_borel_measurable_def: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'c" shows "set_borel_measurable M A f \<equiv> (\<lambda>x. indicat_real A x *\<^sub>R f x) \<in> borel_measurable M"
cases: fixes x :: "tcolor \<times> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" obtains c :: "tcolor" where "x = (c__, \<langle>\<rangle>)"   | c :: "tcolor" and l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and a :: "'c \<times> 'd" and uu_ :: "tcolor" and mkp :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = (c, \<langle>l, (a, uu_, mkp), r\<rangle>)"
bounded_linear_inner_left: fixes b :: "'d" shows "bounded_linear (\<lambda>a. a \<bullet> b)"
postdominate_path_targetnode: fixes n' :: "'node"   and n :: "'node"   and as :: "'edge list"   and n'' :: "'node" assumes "n' postdominates n"   and "n -as\<rightarrow>* n''"   and "n' \<notin> set (sourcenodes as)" shows "n' postdominates n''"
simps_4: fixes c :: "poly"   and n :: "nat"   and p :: "poly"   and c' :: "poly"   and n' :: "nat"   and p' :: "poly" shows "CN c n p *\<^sub>p CN c' n' p' = (if n < n' then CN (c *\<^sub>p CN c' n' p') n (p *\<^sub>p CN c' n' p') else if n' < n then CN (CN c n p *\<^sub>p c') n' (CN c n p *\<^sub>p p') else CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n' (CN c n p *\<^sub>p p'))"
Rep_uexpr_cases: fixes y :: "'e \<Rightarrow> 'f" assumes "y \<in> UNIV" obtains x :: "('f, 'e) uexpr" where "y = \<lbrakk>x\<rbrakk>\<^sub>e"
log_def: fixes a :: "real"   and x :: "real" shows "log a x = ln x / ln a"
equiv_sub_distrib: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u"   and "sub_distrib t" shows "sub_distrib u"
rank_nullity_theorem: fixes scaleC :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "('b, 'c) vec \<Rightarrow> 'd" assumes "linear (*s) scaleC f" shows "finite_dimensional_vector_space.dimension cart_basis = vec.dim {x. f x = (0::'d)} + vector_space.dim scaleC (range f)"
star_fps_rep_Nil: fixes f :: "'d list \<Rightarrow> 'c" shows "star_fps_rep f [] = f []\<^sup>\<star>"
name_swap_bij: fixes a :: "name"   and b :: "name"   and x :: "'b" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
igWlsAllDisj_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igWlsAllDisj MOD \<equiv> igWlsDisj MOD \<and> igWlsAbsDisj MOD"
simps_10: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "not\<^sub>n (\<phi> W\<^sub>n \<psi>) = not\<^sub>n \<phi> M\<^sub>n not\<^sub>n \<psi>"
intros_10: fixes F :: "'b formula" shows "(\<^bold>\<not> (F::'b formula) \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F \<in> AX10"
weakenS: fixes s :: "'a"   and s' :: "'a"   and a :: "'f" assumes "S s s' a" shows "S' s s' a"
typeof_i64: fixes v :: "v" assumes "typeof v = T_i64" shows "\<exists>c. v = ConstInt64 c"
trans: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "transitive A less_eq"   and "less_eq x y"   and "less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
dir: shows "dir = (if \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i then Positive else Negative)"
has_vector_derivative_If_within_closures: fixes x :: "real"   and S :: "real set"   and T :: "real set"   and u :: "real set"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a" assumes "x \<in> S \<union> T"   and "u = S \<union> T"   and "x \<in> S \<union> closure S \<inter> closure T \<Longrightarrow> (f has_vector_derivative f' x) (at x within S \<union> closure S \<inter> closure T)"   and "x \<in> T \<union> closure S \<inter> closure T \<Longrightarrow> (g has_vector_derivative g' x) (at x within T \<union> closure S \<inter> closure T)"   and "\<lbrakk>x \<in> closure S; x \<in> closure T\<rbrakk> \<Longrightarrow> f x = g x"   and "\<lbrakk>x \<in> closure S; x \<in> closure T\<rbrakk> \<Longrightarrow> f' x = g' x" shows "((\<lambda>x. if x \<in> S then f x else g x) has_vector_derivative (if x \<in> S then f' x else g' x)) (at x within u)"
edge_as_set_inj_on_Mi: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and i :: "nat"   and L :: "'b \<Rightarrow> nat" assumes "matching V E M" shows "inj_on edge_as_set (matching_i i V E M L)"
assms: fixes x :: "'a" assumes "x \<in> box a b" shows "g x = f x"
swap_id_eq: fixes a :: "'c"   and b :: "'c"   and x :: "'c" shows "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
iso_class_elems_isomorphic: fixes F :: "('a, 'a set) Maps.arr set"   and f :: "('a, 'a set) Maps.arr"   and f' :: "('a, 'a set) Maps.arr" assumes "Maps.is_iso_class F"   and "f \<in> F"   and "f' \<in> F" shows "Maps.isomorphic f f'"
lunit_ide_simp: fixes f :: "'c" assumes "C.ide f" shows "C'.lunit f = \<r>\<^sub>C[f]"
discrete_topology_unique: fixes U :: "'c set"   and X :: "'c topology" shows "(discrete_topology U = X) = (topspace X = U \<and> (\<forall>x\<in>U. openin X {x}))"
c: shows "h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr) \<rightarrow>\<^sub>r c"
prime_factorization_unique: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "(factorial_semiring.prime_factorization zero times one normalize x = factorial_semiring.prime_factorization zero times one normalize y) = (normalize x = normalize y)"
hom_one: shows "k \<one>\<^bsub>free_Abelian_group T\<^esub> = \<one>\<^bsub>free_Abelian_group S\<^esub>"
inject_2: fixes x2 :: "\<Pi>\<^sub>1 set"   and y2 :: "\<Pi>\<^sub>1 set" shows "(\<alpha>\<nu> x2 = \<alpha>\<nu> y2) = (x2 = y2)"
hg: fixes x :: "'a" assumes "x \<in> S" shows "h x = g x"
ine: shows "interior {x. a \<bullet> x \<le> b} \<noteq> {}"
code: fixes res :: "('a', 'b) resource" shows "local.map_resource res = Resource (map_spmf (map_prod b local.map_resource) \<circ> run_resource res \<circ> a)"
intensional_simps_14: fixes P :: "'b \<Rightarrow> bool" shows "P \<longrightarrow> #False \<equiv> \<not> P"