hcju/isabelleps huggingface dataset:
vrat_plus_def: shows "vrat_plus = (\<lambda>x\<in>\<^sub>\<circ>\<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (rat_of_vrat (x\<lparr>0\<^sub>\<nat>\<rparr>) + rat_of_vrat (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<rat>)"
lfp_F_sup_iterate: shows "lfp F_sup = (\<Squnion>i. (F_sup ^^ i) (\<lambda>x\<in>S. 0))"
simps_1: fixes f :: "'c \<rightarrow> 'b \<rightarrow> 'b"   and d :: "'b" shows "foldr\<cdot>f\<cdot>d\<cdot>[] = d"
simps_2: fixes c :: "'c"   and env :: "'d \<Rightarrow> 'c" shows "eval (Constant c) env = c"
Hermite_of_upt_row_i_in_range_res: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and j :: "'a" assumes "\<not> is_zero_row i A"   and "echelon_form A"   and "ass_function ass"   and "res_function res"   and "mod_type_class.to_nat i < k"   and "k \<le> nrows A"   and "j < i" shows "Hermite_of_upt_row_i A k ass res $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<in> range (res (Hermite_of_upt_row_i A k ass res $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))))"
simps_2: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree" shows "pnet np (p\<^sub>1 \<parallel> p\<^sub>2) = \<lparr>init = {SubnetS s\<^sub>1 s\<^sub>2 |s\<^sub>1 s\<^sub>2. s\<^sub>1 \<in> init (pnet np p\<^sub>1) \<and> s\<^sub>2 \<in> init (pnet np p\<^sub>2)}, trans = pnet_sos (automaton.trans (pnet np p\<^sub>1)) (automaton.trans (pnet np p\<^sub>2))\<rparr>"
set_cover_bound: fixes C :: "nat set"   and C' :: "nat set" assumes "local.inv C {}"   and "sc C' U" shows "sum w C \<le> harm d\<^sup>* * sum w C'"
interiorE: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> interior S" obtains T :: "'c set" where "open T" and "x \<in> T" and "T \<subseteq> S"
rel_fun_mono: fixes X :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and Y :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "(X ===> A) f g"   and "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "(Y ===> B) f g"
prime_elem_gauss_int_iff: fixes z :: "gauss_int" shows "prime_elem z = (prime (gauss_int_norm z) \<or> (\<exists>p. gauss_int_norm z = p\<^sup>2 \<and> prime p \<and> [p = 3] (mod 4)))"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<oplus> Modal M Ms \<Rightarrow>* \<Delta>, m) \<in> derivable (ext R R2 M1 M2) \<longrightarrow> (\<forall>r'\<in>R'. leftPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable (ext R R2 M1 M2))"
simps_4: fixes ux :: "ltkey" shows "catomic (cLtK ux) = True"
simps_2: fixes z :: "complex" shows "ImZ (round_complex z) = round (Im z)"
simps_1: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "ring.coeff R [] = (\<lambda>_. \<zero>\<^bsub>R\<^esub>)"
cart2_append1_between_right_strict: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "z_non_zero p"   and "z_non_zero q"   and "z_non_zero r"   and "B\<^sub>\<real> cart2_pt p cart2_pt q cart2_pt r"   and "q \<noteq> r" shows "\<exists>k\<ge>0. k < 1 \<and> cart2_append1 q = k *\<^sub>R cart2_append1 r + (1 - k) *\<^sub>R cart2_append1 p"
ind_one_str_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c set" assumes "ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U" shows "ind_sheaf.ind_one_str one_str U V \<equiv> one_str (U \<inter> V)"
inc_predict_support_trading_strat': fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and asset :: "'c"   and B :: "'c set" assumes "trading_strategy pf1"   and "asset \<in> support_set pf1 \<union> B" shows "borel_predict_stoch_proc F (pf1 asset)"
freshInp_def: fixes xs :: "'d"   and x :: "'e"   and inp :: "'f \<Rightarrow> ('g, 'h, 'd, 'e, 'i) qTerm set option" shows "freshInp xs x inp \<equiv> liftAll (fresh xs x) inp"
c_failures_last: fixes xps :: "('e \<times> 'f) list"   and X :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" assumes "(xps, X) \<in> c_failures step out s\<^sub>0"   and "xps \<noteq> []" shows "snd (last xps) = out (foldl step s\<^sub>0 (butlast (map fst xps))) (last (map fst xps))"
stk: shows "P,h \<turnstile> stk [:\<le>] ST"
topological_property_of_prod_component: fixes P :: "('c \<times> 'd) topology \<Rightarrow> bool"   and X :: "'c topology"   and Y :: "'d topology"   and Q :: "'c topology \<Rightarrow> bool"   and R :: "'d topology \<Rightarrow> bool" assumes "P (prod_topology X Y)"   and "\<And>x. \<lbrakk>x \<in> topspace X; P (prod_topology X Y)\<rbrakk> \<Longrightarrow> P (subtopology (prod_topology X Y) ({x} \<times> topspace Y))"   and "\<And>y. \<lbrakk>y \<in> topspace Y; P (prod_topology X Y)\<rbrakk> \<Longrightarrow> P (subtopology (prod_topology X Y) (topspace X \<times> {y}))"   and "\<And>X X'. X homeomorphic_space X' \<Longrightarrow> P X = Q X'"   and "\<And>X X'. X homeomorphic_space X' \<Longrightarrow> P X = R X'" shows "topspace (prod_topology X Y) = {} \<or> Q X \<and> R Y"
SuccI3: fixes R :: "('c \<Rightarrow> real) set"   and v :: "'c \<Rightarrow> real"   and t :: "real"   and R' :: "('c \<Rightarrow> real) set" assumes "R \<in> \<R>"   and "v \<in> R"   and "0 \<le> t"   and "v \<oplus> t \<in> R'"   and "R' \<in> \<R>" shows "R' \<in> Succ \<R> R"
sum_gt_neutral_dest': fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "\<one> \<le> a"   and "\<one> < a + b" shows "\<exists>d. Le d \<le> a \<and> Le (- d) \<le> b \<and> (0::'c) \<le> d"
returnU_writer_def: shows "returnU = Writer\<cdot>mempty"
sup_mono: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset"   and d :: "'a multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<union># b \<subseteq># c \<union># d"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) A.map"
inf_le2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) y"
least_map_def: shows "least_map \<equiv> \<lambda>f S. {x. least_mapp f (\<lambda>x. x \<in> S) x}"
eqvt: fixes x :: "lam"   and pi :: "(name \<times> name) list" assumes "NF x" shows "NF (pi \<bullet> x)"
AX100: shows "AX0 \<union> AX10 = AX10"
ereal_open_uminus: fixes S :: "ereal set" assumes "open S" shows "open (uminus ` S)"
bisims1_pc_length_compEs2D: fixes P :: "'a prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and es' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "P,es,h \<turnstile> (es', xs) [\<leftrightarrow>] (stk, loc, length (compEs2 es), xcp)" shows "xcp = None \<and> calls1 es' = None \<and> (\<exists>vs. stk = rev vs \<and> length vs = length es \<and> (is_vals es' \<longrightarrow> es' = map Val vs \<and> xs = loc))"
Trans: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<equiv>\<^sub>e Q"   and "Q \<equiv>\<^sub>e R" shows "P \<equiv>\<^sub>e R"
concat_inf_truncate_nonempty_a: fixes z :: "nat \<Rightarrow> 'd list"   and i :: "nat"   and a :: "'d" assumes "z i = [a]" shows "concat (inf_truncate z i) \<noteq> []"
370: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<sqinter> - z \<le> x \<squnion> y"
assms: fixes s :: "('answer, 'location, 'proc, 'question, 'state) system_state" assumes "P s" shows "\<lbrace>(=) s\<rbrace> c \<lbrace>Q\<rbrace>"
init_model: fixes n :: "nat"   and nid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<nexists>n'. latestAct_cond nid t n n'"   and "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>" shows "bc (\<sigma>\<^bsub>nid\<^esub>t n) = []"
negative_part_def: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.negative_part C = {L. (\<exists>A. L = Neg A) \<and> L \<in> C}"
wpQ_is_pre': fixes c :: "com"   and k :: "nat"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>2\<^sub>' {wp\<^sub>Q c (\<lambda>s. enat k * Q s)} c {\<lambda>s. enat k * Q s}"
IH: fixes i :: "nat"   and j :: "nat" assumes "match test r i j" shows "i \<le> j"
post_weak_hoare_r: fixes p :: "'a upred"   and C :: "'a hrel"   and q\<^sub>2 :: "'a upred"   and q\<^sub>1 :: "'a upred" assumes "\<lbrace>p\<rbrace> C \<lbrace>q\<^sub>2\<rbrace>\<^sub>u"   and "`q\<^sub>2 \<Rightarrow> q\<^sub>1`" shows "\<lbrace>p\<rbrace> C \<lbrace>q\<^sub>1\<rbrace>\<^sub>u"
IH: fixes gpv :: "('a, 'out, 'in) gpv" shows "interaction_bound' (local.restrict_gpv gpv) \<le> interaction_bound' gpv"
case_distrib: fixes h :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and prod :: "'e \<times> 'f" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
mon_of_pol_correct: fixes l :: "'a list"   and P :: "pol"   and M :: "mon" assumes "in_carrier l"   and "mon_of_pol P = Some M" shows "Ipol l P = Imon l M"
P2_security: fixes I :: "('a \<times> 'b) spmf"   and domain :: "'a \<Rightarrow> 'c set"   and range :: "'a \<Rightarrow> 'c set"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "bool \<times> 'a \<times> bool \<times> bool \<Rightarrow> bool spmf"   and B :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and m2 :: "bool"   and HCP_ad :: "real"   and m1 :: "bool \<times> bool" assumes "ETP_base I domain range F F\<^sub>i\<^sub>n\<^sub>v"   and "\<forall>a. lossless_spmf (D a)"   and "\<forall>b\<^sub>\<sigma>. etp.HCP_adv I range F\<^sub>i\<^sub>n\<^sub>v B (ETP_base.\<A> range B) m2 b\<^sub>\<sigma> D \<le> HCP_ad" shows "sim_det_def.adv_P2 (ETP_base.R2 I range B F F\<^sub>i\<^sub>n\<^sub>v) (ETP_base.S2 I range B F\<^sub>i\<^sub>n\<^sub>v) funct_OT_12 m1 m2 D \<le> 2 * HCP_ad"
fds_ln_0: fixes l :: "'b" shows "fds_ln l 0 = fds_const l"
nN_2: fixes p :: "ref"   and no :: "ref"   and not :: "dag" assumes "ordered rt var"   and "Dag p low high rt"   and "Dag no low high not"   and "no \<in> set_of rt" shows "ordered not var"
local_determinism: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and r :: "'a"   and s\<^sub>1 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s\<^sub>2 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s\<^sub>2' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "substitution.revision_step subst r s\<^sub>1 s\<^sub>2"   and "substitution.revision_step subst r s\<^sub>1 s\<^sub>2'" shows "s\<^sub>2 \<approx> s\<^sub>2'"
connectedin_empty: fixes X :: "'c topology" shows "connectedin X {}"
Rice_lm_1: fixes A :: "nat set" assumes "index_set A"   and "A \<noteq> {}"   and "A \<noteq> UNIV"   and "\<exists>n\<in>A. nat_to_ce_set n = {}" shows "one_reducible_to univ_ce (- A)"
in_Ass_not_associated: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "Complete_set_non_associates S"   and "x \<in> S"   and "y \<in> S"   and "x \<noteq> y" shows "normalize x \<noteq> normalize y"
finite_list: fixes A :: "'d set" assumes "finite A" shows "\<exists>xs. set xs = A"
distinct: fixes n :: "nat" shows "State_Monad.ap (State_Monad.return (assert distinct)) (symbols n) = symbols n"
measure_spmf_of_set: fixes S :: "'c set"   and A :: "'c set" shows "Sigma_Algebra.measure (measure_spmf (spmf_of_set S)) A = real (card (S \<inter> A)) / real (card S)"
nxt_wait_least: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream"   and n :: "nat" assumes "ev \<phi> xs"   and "(nxt ^^ n) \<phi> xs" shows "wait \<phi> xs \<le> n"
product_over_sum_right: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y \<and> O x z" shows "P (x \<otimes> y \<oplus> x \<otimes> z) (x \<otimes> (y \<oplus> z))"
fps_mult_nth_conv_upto_subdegree_left: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "fps_nth (f * g) n = (\<Sum>i = subdegree f..n. fps_nth f i * fps_nth g (n - i))"
strip_comb_simps_2: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "unapp t = None" shows "term.strip_comb unapp t = (t, [])"
is_dyn_operand_def: fixes unboxed :: "('a, 'b, 'c) unboxed" shows "is_dyn_operand unboxed = (\<exists>x1. unboxed = OpDyn x1)"
is_invI_2: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" assumes "Group.comm_monoid G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "a \<otimes>\<^bsub>G\<^esub> b = \<one>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> a = b"
perm_supp_eq: fixes p :: "perm"   and x :: "'a" assumes "supp p \<sharp>* x" shows "p \<bullet> x = x"
fA: shows "f (\<Sqinter>A) \<in> carrier L"
simps_4: fixes a :: "Cooper.num" shows "numbound0 (Neg a) = numbound0 a"
that: fixes \<C>' :: "'a set set" assumes "S \<subseteq> \<Union> \<C>'"   and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>'. U \<inter> V \<noteq> {}}" shows "thesis"
prv_imp_cnj3L: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (imp \<phi>1 \<chi>) (imp (cnj \<phi>1 \<phi>2) \<chi>))"
a: shows "\<forall>(a, b)\<in>set (ab # pi). a \<sharp> x \<and> b \<sharp> x"
is_ortho_set_ket: shows "is_ortho_set (range ket)"
pentagon': fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "B.ide k"   and "trg f = src g"   and "trg g = src h"   and "trg h = src k" shows "((k \<star> B.\<a>' f g h) \<cdot> B.\<a>' f (h \<star> g) k) \<cdot> (B.\<a>' g h k \<star> f) = B.\<a>' (g \<star> f) h k \<cdot> B.\<a>' f g (k \<star> h)"
orderI: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a = a \<union># b" shows "b \<subseteq># a"
Eps_Opt_def: fixes P :: "'a \<Rightarrow> bool" shows "Eps_Opt P = (if \<exists>x. P x then Some (SOME x. P x) else None)"
Un_upper_bound: fixes C :: "'a set set" assumes "Chain fTo D C" shows "\<Union> C \<in> upper_bounds fTo D C"
iIN_Suc: fixes x :: "nat"   and n :: "nat"   and d :: "nat" assumes "x \<in> [n\<dots>,d]"   and "x < n + d" shows "Suc x \<in> [n\<dots>,d]"
psubst_new_free: fixes c :: "'c"   and n :: "'c"   and p :: "('c, 'd) form" assumes "c \<noteq> n" shows "new n (psubst (id(n := c)) p)"
Max_ge_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq x (linorder.Max less_eq A) = (\<exists>a\<in>A. less_eq x a)"
star_of_le: fixes x :: "'a"   and y :: "'a" shows "(star_of x \<le> star_of y) = (x \<le> y)"
oalist_inv_list_of_oalist: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalist :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_list :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and x :: "'c" assumes "oalist_abstract rep_key_order list_of_oalist oalist_of_list" shows "oalist_raw.oalist_inv rep_key_order (list_of_oalist x)"
iso_def: shows "iso = \<lparr>iso_verts = id, iso_arcs = arc_to_ends G, iso_head = snd, iso_tail = fst\<rparr>"
funionE: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A |\<union>| B" obtains "c |\<in>| A"   | "c |\<in>| B"
fun_sum_commute_canc: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "\<And>x y. f (x + y) = f x + f y" shows "f (sum g A) = (\<Sum>a\<in>A. f (g a))"
preorder_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "preorder_on A r \<equiv> refl_on A r \<and> trans r"
Rep_frame: fixes x :: "'d frame" shows "Rep_frame x \<in> Collect frame_Rep_set"
partial_corr: shows "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
succ_inject: fixes m :: "'a"   and n :: "'a" shows "(succ m = succ n) = (m = n)"
remove_sort: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and e :: "'b" shows "sortedTree h t \<longrightarrow> sortedTree h (remove h e t)"
\<tau>move2_WhileCondD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and c :: "'addr expr1"   and b :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (while (c) b) pc xcp"   and "pc < length (compE2 c)" shows "\<tau>move2 P h stk c pc xcp"
listset_singletonI: fixes a :: "'a"   and A :: "'a set"   and ys :: "'a list" assumes "a \<in> A"   and "ys = [a]" shows "ys \<in> listset [A]"
arg_def: fixes tm :: "('a, 'b) tm" shows "arg tm = (case tm of Hd \<zeta> \<Rightarrow> Hd \<zeta> | App x21 x22 \<Rightarrow> x22)"
ACI_trans: fixes r :: "'b rexp"   and s :: "'b rexp"   and t :: "'b rexp" assumes "ACI r s"   and "ACI s t" shows "ACI r t"
neg_le_0_iff_le: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) zero = less_eq zero a"
pairwise_coprime_list_def: fixes qs :: "rat poly list" shows "pairwise_coprime_list qs = (\<forall>m<length qs. \<forall>n<length qs. m \<noteq> n \<longrightarrow> comm_monoid_mult_class.coprime (qs ! n) (qs ! m))"
nn_integral_measure_pmf_finite: fixes M :: "'d pmf"   and f :: "'d \<Rightarrow> ennreal" assumes "finite (set_pmf M)"   and "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" shows "integral\<^sup>N (measure_pmf M) f = (\<Sum>x\<in>set_pmf M. f x * ennreal (pmf M x))"
fsupp: fixes s :: "'s" shows "finite (supp (p s))"
PreCoxeterSystemWithDeletion: shows "PreCoxeterSystemWithDeletion (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
iT_Mult_iUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<U> t2 I \<otimes> k. Q t2) = (P (t1 * k). t1 \<U> t2 I. Q (t2 * k))"
eucl_less_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "euclidean_space.eucl_less inner Basis a b = (\<forall>i\<in>Basis. inner a i < inner b i)"
le_iff_le_annos: fixes c1 :: "'a acom"   and c2 :: "'a acom" shows "(c1 \<sqsubseteq> c2) = ((annos c1, annos c2) \<in> listrel {(x, y). x \<sqsubseteq> y} \<and> strip c1 = strip c2)"
subset_basis_in_imp_indep_in: fixes \<E> :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> X"   and "Y \<subseteq> X" shows "indep_in \<E> Y"
box_empty_imp: fixes a :: "real"   and b :: "real" assumes "box a b = {}" shows "g (cbox a b) = z"
D_def: fixes x :: "real" shows "D x = T x - ln (x + 1) + ln x"
symmetric_mpoly_prod: fixes B :: "'a set"   and A :: "nat set"   and f :: "'a \<Rightarrow> 'b mpoly" assumes "\<And>i. i \<in> B \<Longrightarrow> symmetric_mpoly A (f i)" shows "symmetric_mpoly A (prod f B)"
in_span_listE: fixes v :: "rat vec"   and vs :: "rat vec list" assumes "v \<in> gs.span_list vs" obtains c :: "nat \<Rightarrow> rat" where "True" and "v = gs.lincomb_list c vs"
simps_1: fixes l :: "linear_poly"   and r :: "'a" shows "zero_satisfies (LEQ_ns l r) = ((0::'a) \<le> r)"
is_extensional: fixes f :: "'j" assumes "\<not> \<chi>.J.arr f" shows "Fa.map f = B.null"
std: shows "st.d = 2 ^ Suc (Suc k)"
invar_cons_up: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t' :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.invar (t # bq)"   and "BinomialHeapStruc.rank t' < BinomialHeapStruc.rank t"   and "BinomialHeapStruc.tree_invar t'" shows "BinomialHeapStruc.invar (t' # t # bq)"
splitFace_preserve_final: fixes f :: "face"   and g :: "graph"   and f' :: "face"   and i :: "nat"   and j :: "nat"   and ns :: "nat list" assumes "f \<in> set (finals g)"   and "\<not> final f'" shows "f \<in> set (finals (snd (snd (splitFace g i j f' ns))))"
add3: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' 0 a b \<and> ccw' 0 a c" shows "ccw' 0 a (b + c)"
bisim1Cast: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and T :: "ty" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,Cast T e,h \<turnstile> (Cast T e', xs) \<leftrightarrow> (stk, loc, pc, xcp)"
c_at_d: shows "Interp D \<Turnstile> C"
sorted_last: fixes xs :: "'a list"   and x :: "'a" assumes "sorted xs"   and "x \<in> set xs" shows "x \<le> last xs"
connected_rel_def: fixes n :: "nat"   and S :: "(nat \<times> nat) set" shows "connected_rel n S = {(a, b). has_path n S a b}"
simplex_map: fixes x :: "'a set" assumes "x \<in> X" shows "f ` x \<in> X"
foldl_pair_fun_upd_map_of: fixes U :: "('a \<times> 'b) list"   and F :: "'a \<Rightarrow> 'b" assumes "distinct (map fst U)" shows "foldl pair_fun_upd F U = (\<lambda>k. case map_of U k of None \<Rightarrow> F k | Some v \<Rightarrow> v)"
set_takeWhileD: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "x \<in> list.set (takeWhile P xs)" shows "x \<in> list.set xs \<and> P x"
s_ce_def: shows "s_ce \<equiv> \<lambda>e. s1_1 (comp_by_index e index_of_r_to_l)"
zero: fixes f :: "'a \<Rightarrow> 'b" assumes "Modules.additive f" shows "f (0::'a) = (0::'b)"