hcju/isabelleps huggingface dataset:
simps: fixes a1 :: "'msgid \<times> 'op"   and a2 :: "'msgid \<times> 'op" shows "hb a1 a2 = ((\<exists>m1 i m2. a1 = m1 \<and> a2 = m2 \<and> Broadcast m1 \<sqsubset>\<^sup>i Broadcast m2) \<or> (\<exists>m1 i m2. a1 = m1 \<and> a2 = m2 \<and> Deliver m1 \<sqsubset>\<^sup>i Broadcast m2) \<or> (\<exists>m1 m2 m3. a1 = m1 \<and> a2 = m3 \<and> hb m1 m2 \<and> hb m2 m3))"
P: shows "P \<subseteq> {x \<in> A. f x = x}"
unit_eqI: fixes a :: "'b"   and \<mu> :: "'d" assumes "B.obj a"   and "\<guillemotleft>\<mu> : GoH.map\<^sub>0 a \<Rightarrow>\<^sub>D GoH.map a\<guillemotright>"   and "D.iso \<mu>"   and "\<mu> \<cdot>\<^sub>D \<i>\<^sub>D[GoH.map\<^sub>0 a] = (GoH.map \<i>\<^sub>B[a] \<cdot>\<^sub>D GoH.cmp (a, a)) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<mu>)" shows "\<mu> = GoH.unit a"
dot_scaleR_mult_2: fixes a :: "'a"   and k :: "real"   and b :: "'a" shows "a \<bullet> k *\<^sub>R b = k * (a \<bullet> b)"
eq_vector_fraction_iff: fixes x :: "'b"   and u :: "real"   and v :: "real"   and a :: "'b" shows "(x = (u / v) *\<^sub>R a) = (if v = 0 then x = (0::'b) else v *\<^sub>R x = u *\<^sub>R a)"
min: fixes F' :: "'a set set" assumes "F' \<subset> F" shows "S \<subset> affine hull S \<inter> \<Inter> F'"
is_obj_of_type_alt: shows "is_obj_of_type = is_of_type objT"
nat_pow_pow: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "(x [^] n) [^] m = x [^] (n * m)"
Not: fixes F :: "'effect fs_set"   and x :: "('b, 'pred, 'act, 'effect) formula" assumes "x \<in> \<A>[F]" shows "FL_Formula.Not x \<in> \<A>[F]"
h_nz: shows "\<forall>\<^sub>F x in F. h x \<noteq> 0"
set_conv_list: fixes g :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "prod g (set xs) = prod_list (map g (remdups xs))"
span_add1: fixes K :: "('e, 'f) ring_scheme"   and V :: "('e, 'g, 'h) module_scheme"   and U :: "'g set"   and v :: "'g"   and w :: "'g" assumes "vectorspace K V"   and "U \<subseteq> carrier V"   and "v \<in> module.span K V U"   and "w \<in> module.span K V U" shows "v \<oplus>\<^bsub>V\<^esub> w \<in> module.span K V U"
P'_def: shows "P' \<equiv> \<lambda>x. if x \<in> X ` space M then P x else 0"
denote_density_states: fixes \<rho> :: "complex mat"   and S :: "com" assumes "\<rho> \<in> density_states"   and "well_com S" shows "denote S \<rho> \<in> density_states"
enum_ex_UNIV: fixes P :: "'a \<Rightarrow> bool" shows "enum_ex P = Bex UNIV P"
counts: shows "count_terminals \<alpha> \<le> count_terminals (terminals p)"
free_basis_code: fixes G :: "'b list set" shows "code (\<BB>\<^sub>F G)"
valid_path_finite_in_V: fixes P :: "'a llist"   and n :: "nat" assumes "valid_path P"   and "enat n < llength P" shows "P $ n \<in> V"
in_SINK_restrict_current: fixes x :: "'k"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "x \<in> MFMC_Network.SINK g" shows "x \<in> MFMC_Network.SINK (g \<upharpoonleft> \<Gamma> / f)"
merge_sub: fixes C :: "fterm literal set"   and D :: "fterm literal set"   and lmbd :: "char list \<Rightarrow> fterm"   and C' :: "fterm literal set"   and \<mu> :: "char list \<Rightarrow> fterm"   and D' :: "fterm literal set" assumes "vars\<^sub>l\<^sub>s C \<inter> vars\<^sub>l\<^sub>s D = {}"   and "C \<cdot>\<^sub>l\<^sub>s lmbd = C'"   and "D \<cdot>\<^sub>l\<^sub>s \<mu> = D'" shows "\<exists>\<eta>. C \<cdot>\<^sub>l\<^sub>s \<eta> = C' \<and> D \<cdot>\<^sub>l\<^sub>s \<eta> = D'"
rplusinf_bound0: fixes p :: "fm" assumes "isrlfm p" shows "bound0 (plusinf p)"
assms_10: shows "f holomorphic_on {z \<in> S. k < d \<bullet> z}"
abs_eq: fixes x :: "'b ptrm" shows "Snd (abs_trm x) = abs_trm (PSnd x)"
maddux_3_11_pp: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> - y \<squnion> x \<sqinter> - - y = x"
fag_single_additionTr4: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<forall>s t. s \<in> add_set f {a} n \<and> t \<in> add_set f {\<^sub>i- a} n \<longrightarrow> s \<^sub>f+ t = z"
truncate_down_pos: fixes x :: "real"   and p :: "nat" assumes "0 < x" shows "0 < truncate_down p x"
T_meas: shows "Tinv \<in> M \<rightarrow>\<^sub>M M"
rdvd_reduce: fixes g :: "int"   and d :: "int"   and c :: "int"   and t :: "real" assumes "g dvd d"   and "g dvd c"   and "0 < g" shows "(real_of_int d rdvd real_of_int c * t) = (real_of_int (d div g) rdvd real_of_int (c div g) * t)"
Gauss_Jordan_not_0: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "A \<noteq> 0" shows "Gauss_Jordan A \<noteq> 0"
Re_complex_div_lemma: fixes z :: "complex" shows "(0 < Re ((1 - \<i> * z) / (1 + \<i> * z))) = (cmod z < 1)"
range_disjointed_sets': fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M" shows "range (disjointed A) \<subseteq> sets M"
hmem_Sup_Ords: fixes A :: "hf" assumes "A \<noteq> 0"   and "\<And>i. i \<^bold>\<in> A \<Longrightarrow> Ord i" shows "\<Squnion>A \<^bold>\<in> A"
g_def: shows "g = (\<lambda>x. insertion f (isolate_variable_sparse p v x) * f v ^ x)"
th: shows "\<forall>x. Ifm (x # bs) (minusinf p)"
asym: fixes b :: "'a"   and a :: "'a" assumes "lt b a"   and "lt a b" shows "False"
Block_\<tau>execI_Some: 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 V :: "nat"   and T :: "ty"   and v :: "'addr val" assumes "\<tau>exec_move ci P t e h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t {V:T=\<lfloor>v\<rfloor>; e} h (stk, loc, Suc (Suc pc), xcp) (stk', loc', Suc (Suc pc'), xcp')"
FrameStatImpClosed: fixes F :: "'b frame"   and G :: "'b frame"   and p :: "(name \<times> name) list" assumes "F \<hookrightarrow>\<^sub>F G" shows "p \<bullet> F \<hookrightarrow>\<^sub>F p \<bullet> G"
one_chain_line_integral_rec_join: fixes l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "set l = one_chain"   and "distinct l"   and "valid_chain_list l"   and "boundary_chain one_chain"   and "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"   and "finite basis" shows "line_integral F basis (rec_join l) = one_chain_line_integral F basis one_chain"
subst2Chain: fixes yvec :: "name list"   and M :: "'a"   and xvec :: "name list"   and Tvec :: "'b list" assumes "yvec \<sharp>* M"   and "yvec \<sharp>* xvec"   and "yvec \<sharp>* Tvec" shows "yvec \<sharp>* M[xvec::=Tvec]"
card_roots_unity: fixes n :: "nat" assumes "1 \<le> n" shows "card {z. z ^ n = (1::'a)} \<le> n"
r\<^sub>\<UU>_def: shows "r\<^sub>\<UU> = run\<^sub>t (\<delta>\<^sub>\<UU> \<Sigma>) (\<iota>\<^sub>\<UU> \<phi>) w"
Lim_transform_within: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and x :: "'b"   and S :: "'b set"   and d :: "real"   and g :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> l) (at x within S)"   and "0 < d"   and "\<And>x'. \<lbrakk>x' \<in> S; 0 < dist x' x; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "(g \<longlongrightarrow> l) (at x within S)"
Mp_ident_iff: fixes m :: "int"   and f :: "int poly" assumes "poly_mod_2 m" shows "(poly_mod.Mp m f = f) = (\<forall>n. poly.coeff f n \<in> {0..<m})"
conjugate_unit_disc_compl: shows "conjugate ` unit_disc_compl = unit_disc_compl"
distinct_1: fixes x2 :: "'d \<Rightarrow> 'd" shows "Skip \<noteq> Basic x2"
Class_invertible: fixes a :: "'a" assumes "a \<in> G" shows "quotient.invertible (Class a)"
offs_base_zero: fixes i :: "nat"   and ns :: "nat list"   and k :: "nat" assumes "i < length ns" shows "offs ns k ! i = offs ns 0 ! i + k"
hom_injI: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h"   and "\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>" shows "inj_on h (carrier G)"
hnr_RECT: fixes Rx :: "'c \<Rightarrow> 'a \<Rightarrow> assn"   and F :: "assn"   and F' :: "'c \<Rightarrow> 'a \<Rightarrow> assn"   and Ry :: "'d \<Rightarrow> 'b \<Rightarrow> assn"   and cB :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a \<Rightarrow> 'b Heap"   and aB :: "('c \<Rightarrow> 'd nres) \<Rightarrow> 'c \<Rightarrow> 'd nres"   and ax :: "'c"   and px :: "'a" assumes "\<And>cf af ax px. (\<And>ax px. hn_refine (hn_ctxt Rx ax px * F) (cf px) (F' ax px) Ry (af ax)) \<Longrightarrow> hn_refine (hn_ctxt Rx ax px * F) (cB cf px) (F' ax px) Ry (aB af ax)"   and "\<And>x. mono_Heap (\<lambda>f. cB f x)" shows "hn_refine (hn_ctxt Rx ax px * F) (heap.fixp_fun cB px) (F' ax px) Ry (REC\<^sub>T aB ax)"
intro: fixes c :: "nat \<times> nat \<Rightarrow> int"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> int"   and N :: "nat" assumes "RGraph c s t cf"   and "Network_Impl c s t N" shows "RGraph_Impl c s t N cf"
in_set_member: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> ran\<^sub>l xs) = List.member xs x"
has_field_derivative_iff: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b"   and S :: "'b set" shows "(f has_field_derivative D) (at x within S) = ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
emeasure_real: fixes A :: "'a set" shows "\<exists>r\<ge>0. emeasure P A = ennreal r"
Suc_length_conv: fixes n :: "nat"   and xs :: "'g list" shows "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
S_sym: shows "order.greater_eq (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (uminus ` (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}))"
finite_int_segment: fixes a :: "'b"   and b :: "'b" shows "finite {x \<in> \<int>. a \<le> x \<and> x \<le> b}"
l_def: shows "l \<equiv> limsup (\<lambda>n. ereal (real n * (f n / f (Suc n) - 1)))"
image_domain_eq: fixes c :: "('d, 'c) chart" shows "apply_chart c ` domain c = codomain c"
Conjunctive_def: shows "Conjunctive = {x. \<forall>X. times_abc x (Inf_b X) = Inf_c (times_abc x ` X)}"
forward_ix_def: fixes \<O> :: "'c \<Rightarrow> 'd pmf"   and \<K> :: "'c \<Rightarrow> 'c pmf"   and state_list :: "'c list"   and os :: "'d list" shows "HMM3_defs.forward_ix \<O> \<K> state_list os = HMM3_defs.forward_ix_rec \<O> \<K> state_list (IArray os)"
simps_2: fixes at :: "('e, 'f) atm" shows "varsL (Neg at) = varsA at"
decr: fixes p :: "fm"   and x :: "real"   and bs :: "real list" assumes "bound0 p" shows "Ifm (x # bs) p = Ifm bs (decr p)"
enc_stream_enc: fixes I :: "(nat + nat set) list"   and I' :: "(nat + nat set) list"   and w :: "'a list"   and w' :: "'a list" assumes "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "\<forall>x\<in>set I'. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "enc (w, I) = enc (w', I')" shows "stream_enc (w, I) = stream_enc (w', I')"
part_singlPart: fixes I :: "'a set" shows "part I (singlPart I)"
None_notin_image_Some: fixes A :: "'c set" shows "None \<notin> Some ` A"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(Conclusion x2 = Conclusion y2) = (x2 = y2)"
routing_port_ranges_sound: fixes e :: "char list \<times> 'a wordinterval"   and tbl :: "'a routing_rule list"   and s :: "'a wordinterval"   and k :: "'a word" assumes "e \<in> set (routing_port_ranges tbl s)"   and "k \<in> wordinterval_to_set (snd e)"   and "valid_prefixes tbl" shows "fst e = output_iface (routing_table_semantics tbl k)"
not_deleted: shows "to_woot_char m = to_woot_character M m"
assms_3: shows "InvariantVarsF (getF state) F0 Vbl"
collapse: fixes prod :: "'r \<times> 's" shows "(prod.fst prod, prod.snd prod) = prod"
restrict_def: fixes G :: "('c, 'd) labeled_graph" shows "restrict G = LG {(l, v1, v2). (l, v1, v2) \<in> edges G \<and> v1 \<in> vertices G \<and> v2 \<in> vertices G} (vertices G)"
polynomial_incl: fixes K :: "'a set"   and p :: "'a list" assumes "polynomial K p" shows "set p \<subseteq> K"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<longrightarrow>\<^sub>\<beta> x2"   and "\<And>t1 t2 s y. \<lbrakk> t1 \<longrightarrow>\<^sub>\<beta> t2; \<And>z. P z t1 t2\<rbrakk> \<Longrightarrow> P y (App t1 s) (App t2 s)"   and "\<And>s1 s2 t y. \<lbrakk> s1 \<longrightarrow>\<^sub>\<beta> s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App t s1) (App t s2)"   and "\<And>t1 t2 x y. \<lbrakk>x \<sharp> y;  t1 \<longrightarrow>\<^sub>\<beta> t2; \<And>z. P z t1 t2\<rbrakk> \<Longrightarrow> P y Lam [x].t1 Lam [x].t2"   and "\<And>x s t y. \<lbrakk>x \<sharp> y; x \<sharp> s\<rbrakk> \<Longrightarrow> P y (App Lam [x].t s) t[x::=s]" shows "P z x1 x2"
ag_eq_sol2: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and x :: "'c"   and b :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "x \<in> carrier A"   and "b \<in> carrier A"   and "x \<plusminus>\<^bsub>A\<^esub> a = b" shows "x = b \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> a"
indep_def: fixes V :: "'b set"   and E :: "'b set set" shows "indep V E = (\<forall>v\<in>V. \<forall>w\<in>V. v \<noteq> w \<longrightarrow> {v, w} \<notin> E)"
blind_embed_path_same_hash: fixes h :: "'b \<Rightarrow> 'a"   and e :: "'c \<Rightarrow> 'b"   and p :: "('c \<times> 'c rose_tree list \<times> 'c rose_tree list) list"   and t :: "('b, 'a) rose_tree\<^sub>m" shows "hash_tree h (tree_of_zipper\<^sub>m (blind_path e h p, t)) = hash_tree h (tree_of_zipper\<^sub>m (embed_path e p, t))"
my_wf_induct_triple: fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set"   and r3 :: "('c \<times> 'c) set"   and P :: "'a \<times> 'b \<times> 'c \<Rightarrow> bool"   and x :: "'a \<times> 'b \<times> 'c" assumes "wf (r1 <*lex*> r2 <*lex*> r3)"   and "\<And>x. (\<And>y. ((fst y, fst (snd y), snd (snd y)), fst x, fst (snd x), snd (snd x)) \<in> r1 <*lex*> r2 <*lex*> r3 \<longrightarrow> P y) \<Longrightarrow> P x" shows "P x"
Seed_holds_faces_distinct: fixes p :: "nat" shows "faces_distinct Seed\<^bsub>p\<^esub>"
load_word_mem_mod_cache: fixes s :: "'b sparc_state"   and addr :: "word32"   and asi :: "word8"   and new_cache :: "cpu_cache" shows "load_word_mem s addr asi = load_word_mem (s\<lparr>cache := new_cache\<rparr>) addr asi"
unique_factorization_m_alt_def: fixes m :: "int"   and f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "poly_mod.unique_factorization_m m f cfs = (poly_mod.factorization_m m f cfs \<and> (\<forall>dgs. poly_mod.factorization_m m f dgs \<longrightarrow> poly_mod.Mf m dgs = poly_mod.Mf m cfs))"
assms_3: shows "\<forall>r\<in>range Y \<inter> space N. \<exists>A\<in>sets N. range Y \<inter> A = {r}"
assms_8: shows "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> Xin i \<inter> Xin j = {}"
asymptotics_aux_1: fixes x :: "real"   and i :: "nat"   and s :: "real" assumes "x\<^sub>1 \<le> x"   and "i < k"   and "s \<equiv> if 0 \<le> p then 1 else - 1" shows "(bs ! i * x - s * hb * x * ln x powr - (1 + e)) powr p \<le> (bs ! i * x + (hs ! i) x) powr p"
sin_conv_pre_sin: fixes x :: "real" shows "sin x = x * powser (msllist_of_msstream sin_series_stream) (x\<^sup>2)"
x_step: shows "q' \<in> succ M (x, y) q"
ALT: shows "L1 - L2 = L1 \<inter> (ranked_trees A - L2)"
pi1: shows "\<pi> ` b1 \<in># \<B>'"
contract_imp_parcontract: fixes x :: "comb"   and y :: "comb" assumes "x \<rightarrow>\<^sup>1 y" shows "x \<Rrightarrow>\<^sup>1 y"
cases: fixes x :: "('d, 'c) ipurge_rec" obtains I :: "('c \<times> 'c) set" and D :: "'d \<Rightarrow> 'c" and u :: "'c" and x :: "'d" and xs :: "'d list" and ys :: "'d list" where "xa__ = \<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr>"   | v :: "('c \<times> 'c) set" and va :: "'d \<Rightarrow> 'c" and vb :: "'c" and vd :: "'d list" and ve :: "unit" where "xa__ = \<lparr>Pol = v, Map = va, Dom = vb, In = [], Out = vd, \<dots> = ve\<rparr>"
w_1: shows "Finite_Cartesian_Product.transpose A *v w = sr *s w"
u_in_\<X>: shows "u \<in> \<X> k"
subspace_def: fixes K :: "('b, 'c) ring_scheme"   and W :: "'d set"   and V :: "('b, 'd, 'e) module_scheme" shows "subspace K W V \<equiv> vectorspace K V \<and> LinearCombinations.submodule K W V"
iprefixE: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" assumes "xs \<sqsubseteq> f" obtains g :: "nat \<Rightarrow> 'b" where "f = xs \<frown> g"
"1_4": shows "lproject visible (llist_of_stream v) = lproject visible (llist_of_stream w)"
shift_snth_ge: fixes xs :: "'c list"   and p :: "nat"   and s :: "'c stream" assumes "length xs \<le> p" shows "(xs @- s) !! p = s !! (p - length xs)"
hyps: assumes "mat (row_length (M1 @ M2)) (length (M1 @ M2)) (M1 @ M2)" shows "mat (row_length M2) (length M2) M2"
Reds_dom_lcl: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>" shows "dom l' \<subseteq> dom l \<union> fv e"
MST_validn_lower: fixes K :: "nat"   and L :: "nat" assumes "MST_validn K"   and "L \<le> K" shows "MST_validn L"
unlabel_subst: fixes S :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "unlabel S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta> = unlabel (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
D_3: shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (state_measure (V \<union> V') \<Gamma> \<Otimes>\<^sub>M count_space (range BoolVal))"
codomain_compose: fixes a :: "wall" assumes "codomain_wall a = 0 \<and> is_tangle_diagram a" shows "a ~ a \<circ> basic []"
una_Top: fixes P :: "indi \<Rightarrow> (alt \<times> alt) set"   and S :: "alt set" assumes "P \<in> Prof"   and "S \<noteq> {}" shows "f (Top S \<circ> P) \<in> S"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : p\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : p\<^sub>1 \<star> w \<Rightarrow> p\<^sub>1 \<star> w'\<guillemotright>"   and "((k\<^sup>* \<star> h) \<star> \<theta>) \<cdot> \<a>[k\<^sup>* \<star> h, p\<^sub>0, w] \<cdot> (tab \<star> w) = (((k\<^sup>* \<star> h) \<star> \<theta>') \<cdot> \<a>[k\<^sup>* \<star> h, p\<^sub>0, w'] \<cdot> (tab \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<beta> = p\<^sub>1 \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (p\<^sub>0 \<star> \<gamma>)"
inv_start: shows "inv_locate_a (as, lm) (n, la, ra) ires"
llength_lappend_end'E: fixes w :: "'b llist"   and n :: "nat" assumes "llength w = enat (Suc n)" obtains a :: "'b" and v :: "'b llist" where "w = v $ <a>" and "llength v = enat n"
Rep_inverse: fixes x :: "'a Word.word" shows "word_of_int (sint x) = x"
outside_singleton: fixes x :: "'a" shows "outside {x} = - {x}"
characteriseSeq: fixes C :: "'a sequent" shows "\<exists>A B. C = ( A \<Rightarrow>* B)"
cInf_less_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and y :: "'a" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_below less_eq X" shows "less (Inf X) y = (\<exists>x\<in>X. less x y)"
symmetric_complement_closed: fixes x :: "'a" assumes "symmetric x" shows "symmetric (- x)"
tickFree_Cons: fixes a :: "'c event"   and t :: "'c event list" shows "tickFree (a # t) = (a \<noteq> tick \<and> tickFree t)"
intro: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a"   and B :: "'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list" assumes "lbv A r f T B"   and "lbvc_axioms A r T B step \<tau>s" shows "lbvc A r f T B step \<tau>s"
trans_from_sterms': fixes \<xi> :: "'e"   and p' :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and p :: "('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, p'), a, \<xi>', q) \<in> seqp_sos \<Gamma>"   and "wellformed \<Gamma>"   and "p' \<in> sterms \<Gamma> p" shows "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"
local_order_carrier_closed: fixes history :: "nat \<Rightarrow> 'c list"   and e1 :: "'c"   and i :: "nat"   and e2 :: "'c" assumes "node_histories history"   and "node_histories.history_order history e1 i e2" shows "{e1, e2} \<subseteq> set (history i)"
less_INF_D: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 y :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and i :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less y (Inf (f ` A))"   and "i \<in> A" shows "less y (f i)"
oo_2: shows "Ord (\<omega> \<up> \<alpha>2)"
preserves_reflects_arr: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and f :: "'c" assumes "functor A B F" shows "partial_magma.arr B (F f) = partial_magma.arr A f"
31: shows "subst (instInp Pinp (Var xx)) \<langle>\<phi>\<rangle> xx = instInp Pinp \<langle>\<phi>\<rangle>"
value_f_i: fixes i :: "nat" shows "value_flow \<Delta> (f_i i) = value_flow \<Delta> (f_aux i)"
CLIM_CRLIM_iff: fixes f :: "'a \<Rightarrow> complex"   and L :: "complex"   and x :: "'a" shows "f \<midarrow>x\<rightarrow> L = (\<lambda>y. cmod (f y - L)) \<midarrow>x\<rightarrow> 0"
assms: shows "h \<in> set (pdata_pairs_to_list ps)"
c: shows "extreme {q \<in> A. f q \<sim> q \<or> f q = q} (\<lambda>x y. y \<sqsubseteq> x) c"
exec_restrict: shows "\<Gamma>|\<^bsub>A\<^esub>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
square_free_factorization_int_def: fixes f :: "int poly" shows "square_free_factorization_int f = (case x_split f of (n, g) \<Rightarrow> case square_free_factorization_int' g of (d, fs) \<Rightarrow> if n = 0 then (d, fs) else (d, (monom 1 1, n - 1) # fs))"