hcju/isabelleps huggingface dataset:
inf_mono_left: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and a :: "'b::type"   and b :: "'b::type"   and c :: "'b::type" assumes "class.refinement_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (b::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (c::'b::type)) (inf (b::'b::type) c)"
bind_map_gpv: fixes f :: "'f \<Rightarrow> 'e"   and gpv :: "('f, 'c, 'd) gpv"   and g :: "'e \<Rightarrow> ('b, 'c, 'd) gpv" shows "map_gpv f id gpv \<bind> g = gpv \<bind> g \<circ> f"
constant_functor_axioms_def: fixes B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b" shows "constant_functor_axioms B b \<equiv> partial_magma.ide B b"
converse_iff: fixes a :: "'c"   and b :: "'d"   and r :: "('d \<times> 'c) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
assms_2: shows "g \<in> borel_measurable (F t)"
lborel_permutations_of_set_bst_of_list: shows "distr (Pi\<^sub>M (set xs) (\<lambda>x. \<U> 0 1)) (tree_sigma (count_space A)) (\<lambda>p. bst_of_list (sort_key p xs)) = distr (random_perm xs) (tree_sigma (count_space A)) bst_of_list"
True_in_image_Bex: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b set" shows "(True \<in> P ` A) = Bex A P"
prod_zero: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "\<exists>a\<in>A. f a = zero" shows "comm_monoid_mult.prod times one f A = zero"
graph_lm_4: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" shows "graph (f \<circ> g) = graph g O graph f"
scale_left_commute: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) vec" shows "a *s (b *s x) = b *s (a *s x)"
init_model: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and n :: "nat"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> (\<exists>n'<n. active nid (t n'))"   and "active nid (t n)" shows "bc (cmp nid (t n)) = []"
SUP_ereal_add: fixes f :: "nat \<Rightarrow> ereal"   and g :: "nat \<Rightarrow> ereal" assumes "incseq f"   and "incseq g"   and "\<And>i. f i \<noteq> - \<infinity>"   and "\<And>i. g i \<noteq> - \<infinity>" shows "(SUP i. f i + g i) = Sup (range f) + Sup (range g)"
res_seq_res': fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat"   and n :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "res_seq s (Suc k) n (Suc k) = const_res (Suc k) (res_seq s k)"
sd_unit_def: shows "sd_unit = (1::'c, \<bottom>)"
before_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and "is" :: "instr list" shows "(P,C,M,pc \<rhd> is) = prefix is (drop pc (instrs_of P C M))"
inv_at_def: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat" shows "local.inv_at p A m \<equiv> \<forall>i<n. p A i m"
leftderives_trans: fixes a :: "symbol list"   and b :: "symbol list"   and c :: "symbol list" assumes "leftderives a b"   and "leftderives b c" shows "leftderives a c"
set_iterator_sng_def: fixes x :: "'b"   and c :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and \<sigma>0 :: "'a" shows "set_iterator_sng x c f \<sigma>0 = (if c \<sigma>0 then f x \<sigma>0 else \<sigma>0)"
src_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "span_vertical_category.src C \<mu> \<equiv> if partial_magma.arr (span_vertical_category.vcomp C) \<mu> then \<lparr>Chn = partial_magma.dom C (partial_magma.cod C (Leg0 (Dom \<mu>))), Dom = \<lparr>Leg0 = partial_magma.cod C (Leg0 (Dom \<mu>)), Leg1 = partial_magma.cod C (Leg0 (Dom \<mu>))\<rparr>, Cod = \<lparr>Leg0 = partial_magma.cod C (Leg0 (Dom \<mu>)), Leg1 = partial_magma.cod C (Leg0 (Dom \<mu>))\<rparr>\<rparr> else partial_magma.null (span_vertical_category.vcomp C)"
MINIM: fixes a' :: "'a" assumes "a' \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> a' \<noteq> b \<and> (b, a') \<in> r" shows "(a, a') \<in> r"
IH: fixes bs :: "'a list" assumes "length as' = length bs" shows "vec_smult \<alpha> (vec_plus as' bs) = vec_plus (vec_smult \<alpha> as') (vec_smult \<alpha> bs)"
sel_3: fixes x21 :: "'b"   and x22 :: "'b llist" shows "ltl (LCons x21 x22) = x22"
distrib_flincomb: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and rs :: "'b list"   and ms :: "'c list" assumes "FGModuleHom G smult V smult' T"   and "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "T (scalar_mult.lincomb (aezfun_scalar_mult.fsmult smult) rs ms) = scalar_mult.lincomb (aezfun_scalar_mult.fsmult smult') rs (map T ms)"
assms: fixes v :: "'t" assumes "v \<in> keys p" shows "component_of_term v \<in> K"
pwd_spy_id_pwd_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_id_pwd n)"
degen_V0_sound: fixes u :: "'Q \<times> nat"   and m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" assumes "u \<in> g_V0 (degeneralize_ext m)" shows "fst u \<in> V0"
subcocycle_def: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle u = ((\<forall>n. integrable M (u n)) \<and> (\<forall>n m x. u (n + m) x \<le> u n x + u m ((T ^^ n) x)))"
degen_impl_aux_alt: fixes ecnv :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'e" shows "degeneralize_ext ecnv = (if num_acc = 0 then \<lparr>g_V = {(q, x). x = 0 \<and> q \<in> V}, g_E = E_of_succ (\<lambda>(q, x). if x = 0 then (\<lambda>q'. (q', 0)) ` succ_of_E E q else {}), g_V0 = (\<lambda>q'. (q', 0)) ` V0, bg_F = {(q, x). x = 0 \<and> q \<in> V}, \<dots> = ecnv G\<rparr> else \<lparr>g_V = {(q, x). x < num_acc \<and> q \<in> V}, g_E = E_of_succ (\<lambda>(q, i). if i < num_acc then let i' = if i \<in> acc q then (i + 1) mod num_acc else i in (\<lambda>q'. (q', i')) ` succ_of_E E q else {}), g_V0 = (\<lambda>q'. (q', 0)) ` V0, bg_F = {(q, x). x = 0 \<and> 0 \<in> acc q}, \<dots> = ecnv G\<rparr>)"
assms_2: shows "c' \<turnstile> ev \<mapsto> c''"
set_rotate: fixes n :: "nat"   and xs :: "'a list" shows "set (rotateLeft n xs) = set xs"
span_redundant: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> local.span S" shows "local.span (insert x S) = local.span S"
WT_restD_rinit: fixes \<I>_adv :: "('p, 'q) \<I>"   and \<I>_usr :: "('r, 's) \<I>"   and I :: "'t \<Rightarrow> bool"   and rest :: "('t, 'u, 'p, 'r, 'q, 's, 't) rest_scheme" assumes "WT_rest \<I>_adv \<I>_usr I rest" shows "I (rinit rest)"
conj_refine: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and c\<^sub>0 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice" assumes "conj_distrib (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)"   and "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (d::'b::refinement_lattice)"   and "(c\<^sub>1::'b::refinement_lattice) \<sqsubseteq> (d::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c\<^sub>0::'b::refinement_lattice) (c\<^sub>1::'b::refinement_lattice) \<sqsubseteq> (d::'b::refinement_lattice)"
IH_1: shows "continuous_on UNIV (blin_frechet (good_interp I) (simple_term \<theta>\<^sub>1))"
cod_inv: fixes f :: "'a" assumes "local.iso f" shows "cod (local.inv f) = local.dom f"
primepow_cases: fixes d :: "nat" shows "primepow d = (primepow_even d \<and> \<not> primepow_odd d \<and> \<not> prime d \<or> \<not> primepow_even d \<and> primepow_odd d \<and> \<not> prime d \<or> \<not> primepow_even d \<and> \<not> primepow_odd d \<and> prime d)"
trancl_pred_sexpD1: fixes a :: "('b, unit) Old_Datatype.node set"   and b :: "('b, unit) Old_Datatype.node set" assumes "(a, b) \<in> pred_sexp\<^sup>+" shows "a \<in> sexp"
orbit_subset: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b" assumes "s \<in> orbit f (f t)" shows "s \<in> orbit f t"
case_optionE: fixes P :: "bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'d option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'d" where "x = Some y" and "Q y"
a: fixes y :: "real" assumes "y \<in> {l..u}" shows "f' y \<le> a"
normal_lift_dnf_qe: fixes qe :: "'a list \<Rightarrow> 'a fm"   and \<phi> :: "'a fm" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<and> anormal a \<Longrightarrow> normal (qe as)"   and "normal \<phi>" shows "normal (lift_dnf_qe qe \<phi>)"
intros_4: fixes u :: "'e \<Rightarrow> 'd"   and c :: "'e"   and d :: "'d" assumes "- u c < d" shows "dbm_entry_val u None (Some c) (Lt d)"
split_face_distinct1': fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list" assumes "pre_split_face oldF ram1 ram2 newVertexList" shows "distinct (vertices (fst (split_face oldF ram1 ram2 newVertexList)))"
sphere_def: fixes x :: "'d"   and e :: "real" shows "sphere x e = {y. dist x y = e}"
G: shows "\<forall>v\<in>sinks_aux I D U (y # ys). (v, u) \<notin> I"
eval_fm_OrdP: fixes e :: "(name, hf) finfun"   and x :: "tm" shows "eval_fm e (OrdP x) = Ord \<lbrakk>x\<rbrakk>e"
singleton_pq: fixes p :: "'a"   and q :: "'a" assumes "point p"   and "point q" shows "singleton (p ; q\<^sup>T)"
lookup_mult_scalar: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (p \<odot> q) v = lookup (p * proj_poly (component_of_term v) q) (pp_of_term v)"
frp_algo_def: fixes T1 :: "('d, 'f) tree_automaton_rec"   and T2 :: "('e, 'f) tree_automaton_rec" shows "frp_algo T1 T2 \<equiv> \<lparr>wa_cond = frp_cond, wa_step = frp_step (ta_rules T1) (ta_rules T2), wa_initial = frp_initial (ta_initial T1) (ta_initial T2), wa_invar = frp_invar T1 T2\<rparr>"
fset_eq_iff: fixes S :: "'b fset"   and T :: "'b fset" shows "(S = T) = (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
Iic_subset_Iio_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less" shows "(ord.atMost less_eq a \<subseteq> ord.lessThan less b) = less a b"
new_syz_sigs_is_syz_sig: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and v :: "'t" assumes "rb_aux_inv (bs, ss, p # ps)"   and "v \<in> set (new_syz_sigs ss bs p)" shows "is_syz_sig dgrad v"
free_tv_fun_def: fixes f :: "'b \<Rightarrow> 'c" shows "free_tv f = (let S = \<lambda>n. typ_of (f (of_nat_class.of_nat n)) in Type.dom S \<union> cod S)"
1: shows "(l, l') \<in> \<langle>Ra\<rangle>list_rel"
Sup_aform_def: fixes X :: "'b \<times> 'b pdevs" shows "Sup_aform X = fst X + Radius X"
param_bind: fixes Ra :: "('b \<times> 'd) set"   and Rb :: "('c \<times> 'e) set" shows "((\<bind>), (\<bind>)) \<in> \<langle>Ra\<rangle>nres_rel \<rightarrow> (Ra \<rightarrow> \<langle>Rb\<rangle>nres_rel) \<rightarrow> \<langle>Rb\<rangle>nres_rel"
Domain_fst: fixes r :: "('a \<times> 'b) set" shows "Domain r = fst ` r"
sets_measure_pmf_count_space: fixes M :: "'b pmf" shows "measure_pmf.events M = sets (count_space UNIV)"
QE_dnf_eval: fixes step :: "nat \<Rightarrow> nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and opt :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and xs :: "real list" assumes "\<And>var amount new L F \<Gamma>. amount \<le> var + 1 \<Longrightarrow> (\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (step amount var L F) (xs @ \<Gamma>))"   and "\<And>xs F. eval (opt F) xs = eval F xs" shows "eval (QE_dnf opt step \<phi>) xs = eval \<phi> xs"
min_key_val_minimal: fixes z :: "'b \<times> 'c"   and xs :: "('b, 'c) oalist_ntm"   and ko :: "'b nat_term_order" assumes "z \<in> set (fst (list_of_oalist_ntm xs))" shows "ko.le (key_order_of_nat_term_order_inv ko) (fst (OAlist_min_key_val_ntm ko xs)) (fst z)"
keys_def: fixes t :: "('c, 'd) rbt" shows "RBT_Impl.keys t = map fst (RBT_Impl.entries t)"
assms_3: shows "(y + x) mod q = (y + x') mod q"
Abs_inverse: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'c" assumes "type_definition Rep Abs A"   and "y \<in> A" shows "Rep (Abs y) = y"
croots3_def: shows "Cubic_Polynomials.croots3 = roots3_generic solve_depressed_cubic_complex"
set_add_comm: fixes I :: "'a set"   and J :: "'a set" assumes "I \<subseteq> carrier R"   and "J \<subseteq> carrier R" shows "I <+> J = J <+> I"
add_diff_inverse: fixes a :: "'q"   and b :: "'q" assumes "a \<le> b" shows "a + (b - a) = b"
mHom_lin: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and N :: "('l, 'i, 'm) Module_scheme"   and m :: "'h"   and f :: "'h \<Rightarrow> 'l"   and a :: "'i" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "f \<in> mHom R M N"   and "a \<in> carrier R" shows "f (a \<cdot>\<^sub>s\<^bsub>M\<^esub> m) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f m"
oexp_def: shows "rst.oexp = {(f, g). f \<in> rst.FINFUNC \<and> g \<in> rst.FINFUNC \<and> ((let m = wo_rel.max_fun_diff (s +o t) f g in (f m, g m) \<in> r) \<or> f = g)}"
less_not_sym: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.preorder less_eq less"   and "less x y" shows "\<not> less y x"
vector_mul_lcancel: fixes a :: "'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "(a *s x = a *s y) = (a = (0::'a) \<or> x = y)"
sum: shows "\<forall>x. (\<Sum>i. X x * indpre i x) = X x * indicat_real (Y -` (A \<inter> range Y) \<inter> space M) x"
map_dual_inv_def: fixes f :: "'d dual \<Rightarrow> 'e dual" shows "map_dual_inv f = \<partial>\<^sup>- \<circ> f \<circ> \<partial>"
H_def: shows "(H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. (c::'a::refinement_lattice) ; x \<sqinter> (d::'a::refinement_lattice))"
of_hypreal_add: fixes x :: "real star"   and y :: "real star" shows "of_hypreal (x + y) = of_hypreal x + of_hypreal y"
wn_cc2: shows "winding_number (c +++ reversepath c2) z = 0"
set_take_subset: fixes n :: "nat"   and xs :: "'b list" shows "set (take n xs) \<subseteq> set xs"
lincomb_Cons_0_imp_in_Span: fixes v :: "'v"   and vs :: "'v list"   and a :: "'f"   and as :: "'f list" assumes "v \<in> V"   and "set vs \<subseteq> V"   and "a \<noteq> (0::'f)"   and "(a # as) \<bullet>\<cdot> (v # vs) = (0::'v)" shows "v \<in> RSpan vs"
less_matrix_def: fixes A :: "('a, 'b, 'c) matrix"   and B :: "('a, 'b, 'c) matrix" shows "A < B \<equiv> A \<le> B \<and> A \<noteq> B"
swap_multiset: fixes i :: "nat"   and l :: "'c list"   and j :: "nat" assumes "i < length l"   and "j < length l" shows "mset (swap l i j) = mset l"
set_child_nodes_def: shows "set_child_nodes \<equiv> l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes"
sum_code: fixes g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "sum g (set xs) = sum_list (map g (remdups xs))"
intros_1: fixes V :: "'f"   and Use :: "'g \<Rightarrow> 'f set"   and n :: "'g"   and E :: "'g"   and X :: "'g"   and H :: "'f set"   and L :: "'f set" assumes "V \<in> Use n" shows "(Node n, V) \<in> lift_Use_set Use E X H L"
finite_digraphE: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "finite_digraph \<alpha> \<CC>" obtains "digraph \<alpha> \<CC>" and "vfinite (\<CC>\<lparr>Obj\<rparr>)" and "vfinite (\<CC>\<lparr>Arr\<rparr>)"
f_Exec_Stream_Init_hd: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "hd (f_Exec_Comp_Stream_Init trans_fun xs c) = c"
scaleR_left_le_one_le: fixes x :: "'a"   and a :: "real" assumes "(0::'a) \<le> x"   and "a \<le> 1" shows "a *\<^sub>R x \<le> x"
card_of_nlists_Succ: fixes A :: "'a set"   and n :: "nat" shows "|nlists A (Suc n)| =o |A \<times> nlists A n|"
ds_monotone: fixes S :: "'a set"   and T :: "'a set"   and r :: "('a \<times> 'a) set" assumes "S \<subseteq> T" shows "r \<down>s S \<subseteq> r \<down>s T"
R7_1: shows "pmf (sds R7) a = 1 / 2"
order_divides_expdiff: fixes n :: "nat"   and a :: "nat"   and d :: "nat"   and e :: "nat" assumes "coprime n a" shows "[a ^ d = a ^ e] (mod n) = [d = e] (mod ord n a)"
lemma_3_5_iii_2: fixes a :: "'a" shows "d2 a (1::'a) = a"
finite_entropy_measurable: fixes S :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real" assumes "finite_entropy S X Px" shows "Px \<in> borel_measurable S"
TableInv_axioms_def: fixes f1 :: "real"   and f2 :: "real"   and e :: "real"   and c :: "real"   and l0 :: "nat" shows "TableInv_axioms f1 f2 e c l0 \<equiv> (1 / (f2 * (e - 1)) \<le> real l0 \<and> 1 / (f1 * (c - 1)) \<le> real l0) \<and> f1 / (f2 - f1 * e) \<le> real l0 \<and> f2 / (f2 - f1 * c) \<le> real l0"
isDiagonal_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "isDiagonal A = (\<forall>a b. mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b \<longrightarrow> A $h a $h b = (0::'b))"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b sort_key"   and b :: "'c sort_key" assumes "rel_sort_key R a b" obtains x1 :: "'b \<times> sort_dir" and x2a :: "'b sort_key" and y1 :: "'c \<times> sort_dir" and y2a :: "'c sort_key" where "a = NonFinal x1 x2a" and "b = NonFinal y1 y2a" and "rel_prod R (=) x1 y1" and "rel_sort_key R x2a y2a"   | x :: "'b" and y :: "'c" where "a = Final x" and "b = Final y" and "R x y"
assms_3: shows "z \<notin> cbox a1 a3"
a2: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>' m. m \<le> n' \<and> ( \<Phi>' + \<Gamma>' \<Rightarrow>* \<Psi>' + \<Delta>', m) \<in> derivable (ext R R2 M1 M2) \<and> p = ( \<Phi>' \<Rightarrow>* \<Psi>' \<oplus> Modal M Ms)"
trcl_empty_cons: fixes c :: "'c"   and c' :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" assumes "(c, [], c') \<in> trcl t" shows "c = c'"
Data_def: fixes L :: "'a list"   and D :: "'a dataspace" shows "Data L D = (length L = PartNum D \<and> (\<forall>i\<in>{n. n < PartNum D}. L ! i \<in> D !D! i))"
mset_size_le1_cases: fixes M :: "'a multiset" assumes "size M \<le> Suc 0" obtains "M = {#}"   | m :: "'a" where "M = {#m#}"
mono_implies_arr: fixes f :: "'s" assumes "local.mono f" shows "arr f"
cond_and_distr: fixes b :: "'a upred"   and P :: "'a upred"   and Q :: "'a upred"   and R :: "'a upred"   and S :: "'a upred" shows "(P \<and> Q) \<triangleleft> b \<triangleright> (R \<and> S) = (P \<triangleleft> b \<triangleright> R \<and> Q \<triangleleft> b \<triangleright> S)"
not_embedS_iso: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "\<not> (embedS r r' f \<and> iso r r' f)"
drop: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "Semantics.matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m action.Drop], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
hom_boundary_carrier: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" shows "hom_boundary p X S c \<in> carrier (homology_group (p - 1) (subtopology X S))"
subst_comp_eq_if_disjoint_vars: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and \<gamma> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> (subst_domain \<gamma> \<union> range_vars \<gamma>) = {}" shows "\<gamma> \<circ>\<^sub>s \<delta> = \<delta> \<circ>\<^sub>s \<gamma>"
frac_eq: fixes x :: "'d" shows "(frac x = x) = ((0::'d) \<le> x \<and> x < (1::'d))"
outgoing_alt: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "Graph.outgoing c u = Pair u ` Graph.E c `` {u}"
homeomorphism_st_proj2: shows "homeomorphism (UNIV - {bot_sphere}) UNIV st_proj2 st_proj2_inv"
st_byte_mask_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and address :: "word32" shows "st_byte_mask instr address \<equiv> if fst instr \<in> {load_store_type STD, load_store_type ST, load_store_type STA, load_store_type STDA} then 15 else if fst instr \<in> {load_store_type STH, load_store_type STHA} then if ucast address = 0 then 12 else 3 else if ucast address = 0 then 8 else if ucast address = 1 then 4 else if ucast address = 2 then 2 else 1"
in_hhomE: fixes \<mu> :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C b\<guillemotright>" obtains "C.arr \<mu>" and "C.obj a" and "C.obj b" and "src\<^sub>C \<mu> = a" and "trg\<^sub>C \<mu> = b"
mHom_Z_M: fixes M :: "('i, 'a, 'j) Module_scheme"   and e :: "'k" assumes "R module M" shows "mHom R (Zm R e) M = {mzeromap (Zm R e) M}"
ex_bool_eq: fixes P :: "bool \<Rightarrow> bool" shows "(\<exists>b. P b) = (P True \<or> P False)"
axioms_2: fixes G :: "('e, 'f) monoid_scheme" assumes "finite_comm_group G" shows "comm_group G"
simps_2: fixes lnk :: "'index \<Rightarrow> 'node \<Rightarrow> 'node"   and lbl :: "'node \<Rightarrow> 'index"   and x :: "'node"   and y :: "'node"   and l :: "'node list" shows "stack lnk lbl x (y # l) = (x \<noteq> nil \<and> x = y \<and> x \<notin> set l \<and> stack lnk lbl (lnk (lbl x) x) l)"
path_decomp': fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "G.path (xs @ x # ys)" shows "G.path (xs @ [x])"
safe_distance_4r_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance_4r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = (v\<^sub>o + a\<^sub>o * \<delta> - v\<^sub>e)\<^sup>2 / 2 / (a\<^sub>o - a\<^sub>e) - v\<^sub>o * \<delta> - 1 / 2 * a\<^sub>o * \<delta>\<^sup>2 + v\<^sub>e * \<delta>"
matchs_match_list_comb: fixes name :: "name"   and xs :: "term list"   and ys :: "'c list" shows "match (name $$ xs) (name $$ ys) = matchs xs ys"
currentD_IN: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and a :: "'k" assumes "current \<Gamma> f"   and "a \<in> A \<Gamma>" shows "d_IN f a = 0"