hcju/isabelleps huggingface dataset:
da_elim_cases_21: fixes Env :: "env"   and B :: "lname set"   and C :: "qtname"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Init C\<rangle>\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
subset_box_imp_1: fixes a :: "'a"   and c :: "'a"   and d :: "'a"   and b :: "'a" assumes "\<forall>i\<in>Basis. a \<bullet> i \<le> c \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i" shows "cbox c d \<subseteq> cbox a b"
eq_equiv_class: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c"   and A :: "'c set" assumes "r `` {a} = r `` {b}"   and "equiv A r"   and "b \<in> A" shows "(a, b) \<in> r"
in_imp_ex_inext_nth: fixes x :: "nat"   and I :: "nat set" assumes "x \<in> I" shows "\<exists>n. x = I \<rightarrow> n"
lnths_def: fixes xs :: "'b llist"   and A :: "nat set" shows "lselect A xs = lmap fst (lfilter (\<lambda>(x, y). y \<in> A) (xs \<bar>\<bar> iterates Suc 0))"
nn_integral_cong_AE: fixes u :: "'b \<Rightarrow> ennreal"   and v :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "AE x in M. u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
filter_min_append_relE: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "transp rel"   and "reflp rel"   and "x \<in> set xs \<union> set ys" obtains y :: "'c" where "y \<in> set (filter_min_append rel xs ys)" and "rel y x"
comp_target_ide: fixes b :: "'a"   and f :: "'a" assumes "b \<in> targets f"   and "ide f" shows "b \<star> f \<cong> f"
get_terms_def: fixes literal :: "'a literal" shows "get_terms literal = (case literal of Pos x11 x12 \<Rightarrow> x12 | Neg x21 x22 \<Rightarrow> x22)"
complete_uniform: fixes "open" :: "'c set \<Rightarrow> bool"   and uniformity :: "('c \<times> 'c) filter"   and S :: "'c set" assumes "class.uniform_space open uniformity" shows "uniform_space.complete open uniformity S = (\<forall>F\<le>principal S. F \<noteq> bot \<longrightarrow> uniform_space.cauchy_filter uniformity F \<longrightarrow> (\<exists>x\<in>S. F \<le> topological_space.nhds open x))"
suminf_def: fixes f :: "nat \<Rightarrow> 'c" shows "suminf f = (THE s. f sums s)"
le': fixes x :: "'b" shows "spmf p x \<le> spmf q x"
LeadsTo_cancel_Diff2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w A' \<union> B"   and "F \<in> B - A' \<longmapsto>w B'" shows "F \<in> A \<longmapsto>w A' \<union> B'"
is_acc_run_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" shows "gb_graph.is_acc_run (b_graph.to_gbg_ext (degeneralize_ext ma) m) r \<equiv> graph_defs.is_run (b_graph.to_gbg_ext (degeneralize_ext ma) m) r \<and> gb_graph.is_acc (b_graph.to_gbg_ext (degeneralize_ext ma) m) r"
less_asym: fixes x :: "'b"   and y :: "'b"   and P :: "bool" assumes "x < y"   and "\<not> P \<Longrightarrow> y < x" shows "P"
insert_Diff_single: fixes a :: "'e"   and A :: "'e set" shows "insert a (A - {a}) = insert a A"
ctr_transfer_1: fixes R :: "'l \<Rightarrow> 'm \<Rightarrow> bool" shows "rel_option R None None"
saction_simps_3: fixes P :: "'a prog"   and a :: "'b"   and al :: "addr_loc"   and v :: "'b val" shows "saction P (NormalAction (WriteMem a al v)) = is_volatile P al"
gcd_of_int_poly: fixes f :: "int poly"   and g :: "int poly" shows "gcd (of_int_poly f) (of_int_poly g) = smult (inverse (of_int (lead_coeff (gcd f g)))) (of_int_poly (gcd f g))"
lcm_gauss_int_def: shows "lcm \<equiv> normalization_euclidean_semiring_class.lcm"
length_lang_pow_lb: fixes A :: "'b list set"   and k :: "nat"   and w :: "'b list"   and n :: "nat" assumes "\<forall>w\<in>A. k \<le> length w"   and "w \<in> A ^^ n" shows "k * n \<le> length w"
invert_side_of_triangle_1: fixes h :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.arr h"   and "f \<odot> g = h"   and "Maps.iso f" shows "Maps.seq (Maps.inv f) h \<and> g = Maps.inv f \<odot> h"
rtrancl_image_advance: fixes q :: "'b"   and R :: "('b \<times> 'b) set"   and Q0 :: "'b set"   and x :: "'b" assumes "q \<in> R\<^sup>* `` Q0"   and "(q, x) \<in> R" shows "x \<in> R\<^sup>* `` Q0"
tendsto_vec_nth: fixes f :: "'a \<Rightarrow> ('b, 'c) vec"   and a :: "('b, 'c) vec"   and net :: "'a filter"   and i :: "'c" assumes "(f \<longlongrightarrow> a) net" shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
g_orbit_eq: fixes X :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> bool"   and U :: "'d set" shows "g_orbit X G U = {X t |t. t \<in> U \<and> (\<forall>\<tau>\<in>down U t. G (X \<tau>))}"
sel_3: fixes x1 :: "symbol \<times> symbol list"   and x2 :: "nat"   and x3 :: "nat"   and x4 :: "nat" shows "item_origin (Item x1 x2 x3 x4) = x3"
pathstart_join: fixes g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" shows "pathstart (g1 +++ g2) = pathstart g1"
assms_3: shows "A \<in> carrier_mat (length fs) (length gs)"
Lim_transform_within: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and x :: "'a"   and S :: "'a set"   and d :: "real"   and g :: "'a \<Rightarrow> 'b" 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)"
arrE: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" obtains "A1.arr (fst f) \<and> A2.arr (snd f)"
lfp_prod: fixes F :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b" assumes "mono F" shows "lfp F = (lfp (\<lambda>x. fst (F (x, lfp (\<lambda>y. snd (F (x, y)))))), lfp (\<lambda>y. snd (F (lfp (\<lambda>x. fst (F (x, lfp (\<lambda>y. snd (F (x, y)))))), y))))"
assert_opt_def: fixes v :: "'d option" shows "assert_opt v \<equiv> case v of None \<Rightarrow> fail | Some v \<Rightarrow> return v"
l_comb_sc1Tr: fixes A :: "'b set"   and H :: "'a set"   and r :: "'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "r \<in> carrier R"   and "H \<noteq> {}" shows "s \<in> {j. j \<le> n} \<rightarrow> A \<and> g \<in> {j. j \<le> n} \<rightarrow> H \<longrightarrow> r \<cdot>\<^sub>s \<Sigma>\<^sub>e M (\<lambda>k. s k \<cdot>\<^sub>s g k) n = \<Sigma>\<^sub>e M (\<lambda>k. (r \<cdot>\<^sub>r\<^bsub>R\<^esub> s k) \<cdot>\<^sub>s g k) n"
min_range_antimono: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and l :: "float"   and u :: "float"   and p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and DF :: "floatarith \<Rightarrow> floatarith"   and Y :: "(real \<times> real pdevs) \<times> real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "x \<in> aform_err e X"   and "real_of_float l \<le> x"   and "x \<le> real_of_float u"   and "min_range_antimono p F DF l u X = Some Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}"   and "\<And>x. real_of_float x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> interpret_floatarith (F (Num x)) [] = f (real_of_float x)"   and "\<And>x. real_of_float x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> interpret_floatarith (DF (Num x)) [] = f' (real_of_float x)"   and "\<And>x. x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> (f has_real_derivative f' x) (at x)"   and "\<And>x. x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> f' x \<le> max (f' (real_of_float l)) (f' (real_of_float u))" shows "f x \<in> aform_err e Y"
HMA_M3: shows "HMA_M3 (P, A, Q) (P', A', Q')"
cases: fixes r :: "'c gorder" obtains carrier :: "'c set" and eqa :: "'c \<Rightarrow> 'c \<Rightarrow> bool" and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" where "r = \<lparr>carrier = carrier, eq = eqa, le = le\<rparr>"
has_measure_limit: fixes S :: "'c set"   and e :: "real" assumes "S \<in> lmeasurable"   and "0 < e" obtains B :: "real" where "0 < B" and "\<And>a b. ball (0::'c) B \<subseteq> cbox a b \<Longrightarrow> \<bar>Sigma_Algebra.measure lebesgue (S \<inter> cbox a b) - Sigma_Algebra.measure lebesgue S\<bar> < e"
ns\<^sub>1_2: shows "set ns\<^sub>1 \<subseteq> set ns"
imgTr8: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and H :: "'e set"   and u :: "'g"   and v :: "'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "F \<guillemotright> H "   and "u \<in> f ` H"   and "v \<in> f ` H" shows "u \<cdot>\<^bsub>G\<^esub> v \<in> f ` H"
v0: shows "v \<noteq> 0\<^sub>v n"
iso_cblinfun_def: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "iso_cblinfun A = (\<exists>B. A o\<^sub>C\<^sub>L B = id_cblinfun \<and> B o\<^sub>C\<^sub>L A = id_cblinfun)"
prefix_genPrefix_trans: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list"   and r :: "('a \<times> 'a) set" assumes "x \<le> y"   and "(y, z) \<in> genPrefix r" shows "(x, z) \<in> genPrefix r"
wls_swap_commute: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and zs' :: "'varSort"   and x :: "'var"   and y :: "'var"   and x' :: "'var"   and y' :: "'var" assumes "wls s X"   and "zs \<noteq> zs' \<or> {x, y} \<inter> {x', y'} = {}" shows "X #[x \<and> y]_zs #[x' \<and> y']_zs' = X #[x' \<and> y']_zs' #[x \<and> y]_zs"
assms: shows "\<forall>i\<in>I. p i =o r i"
quot_to_ratfps_def: shows "local.quot_to_ratfps \<equiv> map_fun id Abs_ratfps (\<lambda>(x, y). let (x', y') = normalize_quot (x, y) in if constant_term y' = (0::'a) then 0 else quot_to_fract (x', y'))"
\<epsilon>o_def: fixes x :: "'b" shows "F'G'.\<epsilon>o x = \<psi>' x (G' x)"
True_steps_concD: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" assumes "(True # p, q) \<in> steps (RegExp2NA.conc L R) w" shows "(\<exists>r. (p, r) \<in> steps L w \<and> q = True # r) \<or> (\<exists>u a v. w = u @ a # v \<and> (\<exists>r. (p, r) \<in> steps L u \<and> fin L r \<and> (\<exists>s. (start R, s) \<in> step R a \<and> (\<exists>t. (s, t) \<in> steps R v \<and> q = False # t))))"
wls_imp_good: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls s X" shows "good X"
tendsto_MInfty: fixes f :: "'a \<Rightarrow> ereal"   and F :: "'a filter" shows "(f \<longlongrightarrow> - \<infinity>) F = (\<forall>r. \<forall>\<^sub>F x in F. f x < ereal r)"
state_at_0_1: fixes i :: "nat" shows "block i 0 = Partial_Recursive.list_encode [i, 0]"
cr_pmf_def: shows "cr_pmf \<equiv> \<lambda>x y. x = measure_pmf y"
unlabel_mem_has_label: fixes x :: "'d"   and A :: "('e \<times> 'd) list" assumes "x \<in> set (unlabel A)" shows "\<exists>l. (l, x) \<in> set A"
induced_modulus: shows "\<exists>d<n. A.induced_modulus d"
splitDigit_list: fixes p :: "'c \<Rightarrow> bool"   and i :: "'c"   and d :: "('b, 'c) Digit"   and l :: "('b, 'c) Node list"   and n :: "('b, 'c) Node"   and r :: "('b, 'c) Node list" assumes "splitDigit p i d = (l, n, r)" shows "digitToList d = nlistToList l @ nodeToList n @ nlistToList r \<and> length l \<le> 4 \<and> length r \<le> 4"
fun_update: fixes S :: "'a st"   and x :: "char list"   and y :: "'a" shows "fun (update S x y) = (fun S)(x := y)"
comp_inv_arr': fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.comp (CC.inv f) f = CC.dom f"
cInf_insert: 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 X :: "'b set"   and a :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_below less_eq X" shows "Inf (insert a X) = inf a (Inf X)"
right_space: fixes ts :: "traffic"   and v :: "view"   and c :: "cars" shows "right (space ts v c) = pos ts c + regular (own v) ts c"
simps_2: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and e :: "'c"   and xs :: "('a \<times> 'c) list"   and q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" shows "ListUpdate f ((x, e) # xs) q = (ListUpdate f xs q)(x := q e x)"
inL1: fixes G :: "'a formula"   and H :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "G, H, \<Gamma> \<Rightarrow> \<Delta>" shows "H, G, \<Gamma> \<Rightarrow> \<Delta>"
one: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" shows "\<one>\<^bsub>G\<^esub> \<in> generate G H"
mset_remdups: fixes xs :: "'b list" shows "mset (remdups xs) = mset_set (set xs)"
subset_singleton_iff: fixes X :: "'d set"   and a :: "'d" shows "(X \<subseteq> {a}) = (X = {} \<or> X = {a})"
hyperb_ineq_4_points': fixes e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and t :: "'a" shows "Min {Gromov_product_at e x y, Gromov_product_at e y z, Gromov_product_at e z t} \<le> Gromov_product_at e x t + 2 * deltaG TYPE('a)"
f_2: fixes x :: "real" assumes "x \<in> {a..b}" shows "0 \<le> f x"
member_rec_2: fixes y :: "'c" shows "List.member [] y = False"
image_mset_def: fixes f :: "'b \<Rightarrow> 'c" shows "image_mset f = fold_mset (add_mset \<circ> f) {#}"
map\<^sub>1_simps_4: fixes f :: "'c" assumes "C.ide f" shows "D.dom (\<sigma>\<^sub>1 f) = G f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)"
len_upshift: fixes P :: "'v" shows "len (upshift P) = (case len P of 0 \<Rightarrow> 0 | Suc nat \<Rightarrow> Suc (Suc nat))"
deg_pm_plus: fixes s :: "'d \<Rightarrow>\<^sub>0 'c"   and t :: "'d \<Rightarrow>\<^sub>0 'c" shows "deg_pm (s + t) = deg_pm s + deg_pm t"
PossiblyNoContingentObjectExistsAxiom: shows "\<exists>w. \<forall>x. ConcreteInWorld x w \<longrightarrow> (\<forall>v. ConcreteInWorld x v)"
UNIV_not_singleton: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.perfect_space open" shows "UNIV \<noteq> {x}"
remove_all_removes: fixes x :: "'b"   and xs :: "'b list" shows "x \<notin> set (remove_all x xs)"
simps_4: fixes va :: "nat"   and uw :: "bdd" shows "Rmi_g (Suc (Suc va)) Trueif uw = False"
preserves_reflects_arr: fixes f :: "'b \<times> 'b" shows "C.arr (\<Phi>\<^sub>F f) = B.VV.arr f"
monotonic: fixes P :: "pi"   and A :: "(pi \<times> pi) set"   and P' :: "pi"   and B :: "(pi \<times> pi) set" assumes "P \<leadsto>\<^sup>^<A> P'"   and "A \<subseteq> B" shows "P \<leadsto>\<^sup>^<B> P'"
fbox_whilei: fixes p :: "('b \<times> 'b) set"   and i :: "('b \<times> 'b) set"   and t :: "('b \<times> 'b) set"   and q :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" assumes "rdom p \<subseteq> rdom i"   and "rdom i ; rel_ad t \<subseteq> rdom q"   and "rdom i ; rdom t \<subseteq> wp x i" shows "rdom p \<subseteq> wp (rel_antidomain_kleene_algebra.whilei t i x) q"
poly_mult_l_distr': fixes p1 :: "'a list"   and p2 :: "'a list"   and p3 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R"   and "set p3 \<subseteq> carrier R" shows "poly_mult (poly_add p1 p2) p3 = poly_add (poly_mult p1 p3) (poly_mult p2 p3)"
add_commute: fixes a :: "('a \<times> ('b \<times> 'a) list) set"   and b :: "('a \<times> ('b \<times> 'a) list) set" shows "a \<union> b = b \<union> a"
ilist_slice_nth: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and m :: "nat" shows "ilist_slice f k m = map f [m * k..<Suc m * k]"
inverse_pow_pow: fixes G :: "('d, 'e) cyclic_group_scheme"   and a :: "'d"   and r :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "a \<in> carrier G" shows "inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> r) = inv\<^bsub>G\<^esub> a [^]\<^bsub>G\<^esub> r"
preserves_iso: fixes f :: "'d \<times> 'd \<times> 'd" assumes "CCC.iso f" shows "local.iso (\<alpha>' f)"
lt_MP_oalist: fixes xs :: "('t, 'c) oalist_ntm" shows "lt (MP_oalist xs) = (if is_zero (MP_oalist xs) then min_term else fst (OAlist_min_key_val_ntm cmp_term xs))"
prod_pOp_mem: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and X :: "'c \<Rightarrow> 'd"   and Y :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A" shows "prod_pOp I A X Y \<in> carr_prodag I A"
sum_dvd: shows "even (\<Sum>bl\<in>#remove1_mset b1 \<B>. b1 |\<inter>| bl * (b1 |\<inter>| bl - 1))"
minus_set_fold: fixes A :: "'g set"   and xs :: "'g list" shows "A - set xs = fold Set.remove xs A"
Ints_abs: 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 a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "a \<in> ring_1.Ints one plus zero minus" shows "abs a \<in> ring_1.Ints one plus zero minus"
ccpo_Sup_least: fixes A :: "'b set"   and z :: "'b" assumes "Complete_Partial_Order.chain ord' A"   and "\<And>x. x \<in> A \<Longrightarrow> ord' x z" shows "ord' (lub' A) z"
InvInfoAtBetaUpright_def: fixes c :: "('c, 'd, 'e) configuration_scheme" shows "InvInfoAtBetaUpright c = (\<forall>k p q. beta_upright (InfoAt c k p q) (IncomingInfo c (k + 1) p q))"
min_add_distrib_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'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"   and z :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus (ord.min less_eq x y) z = ord.min less_eq (plus x z) (plus y z)"
map_of_delete_aux': fixes xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst xs)" shows "map_of (delete_aux k xs) = (map_of xs)(k := None)"
Nodes_subset: fixes i :: "nat"   and levellista :: "ref list list" shows "Nodes i levellista \<subseteq> Nodes (Suc i) levellista"
a: shows "fin (M[a\<turnstile>c>b]) x"
responses_\<I>_full: fixes x :: "'d" shows "responses_\<I> \<I>_full x = UNIV"
RPD_safe: fixes g :: "safety"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Past g (from_mregex mr \<phi>s)"   and "ms \<in> RPD mr" shows "safe_regex Past g (from_mregex ms \<phi>s)"
type_universe_def: fixes t :: "pdf_type" shows "type_universe t = {v. val_type v = t}"
assms_3: shows "g \<in> punit.reduced_GB (monomial (1::'a) ` S)"
ide_MkIde: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a" assumes "concrete_category Obj Hom Id Comp"   and "A \<in> Obj" shows "partial_magma.ide (concrete_category.COMP Obj Hom Comp) (concrete_category.MkArr A A (Id A))"
absorb_iff1: fixes b :: "'d upred"   and a :: "'d upred" shows "(a \<sqsubseteq> b) = ((a \<or> b) = a)"
compatible_SN: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" assumes "SN S"   and "NS O S \<subseteq> S" shows "SN (S O S\<^sup>* O NS\<^sup>*)"
assms_3: shows "(fi, f) \<in> A \<rightarrow> B"
code_unique_dec: fixes \<C> :: "'b list set"   and us :: "'b list list" assumes "code \<C>"   and "us \<in> lists \<C>" shows "Dec \<C> concat us = us"
eucl_rel_int_by0: fixes k :: "int" shows "eucl_rel_int k 0 (0, k)"
hyps_1: shows "c \<noteq> (0::'a) \<or> cs \<noteq> 0"
assoc'_eq_inv_assoc: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = D.inv \<a>\<^sub>D[f, g, h]"
defensive_move_exists_for_Odd: fixes p :: "'a list" assumes "position p" shows "winning_position_Even p \<or> (\<exists>m. move_defensive_by_Odd m p)"
SUP_ereal_mult_left: fixes I :: "'d set"   and f :: "'d \<Rightarrow> ereal"   and c :: "ereal" assumes "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"   and "0 \<le> c" shows "(\<Squnion>i\<in>I. c * f i) = c * \<Squnion> (f ` I)"
ring_inv1: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and b :: "'c" assumes "Ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "-\<^sub>a\<^bsub>R\<^esub> a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = (-\<^sub>a\<^bsub>R\<^esub> a) \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<and> -\<^sub>a\<^bsub>R\<^esub> a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = a \<cdot>\<^sub>r\<^bsub>R\<^esub> (-\<^sub>a\<^bsub>R\<^esub> b)"
enn2real_ennreal: fixes r :: "real" assumes "0 \<le> r" shows "enn2real (ennreal r) = r"
bind_map_spmf: fixes f :: "'t \<Rightarrow> 's"   and p :: "'t spmf"   and g :: "'s \<Rightarrow> 'r spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
span_is_submodule: fixes S :: "'c set" assumes "S \<subseteq> carrier (md N)" shows "LinearCombinations.submodule R (w.span S) (md N)"
hom_mult: fixes x :: "'b poly"   and y :: "'b poly" shows "map_poly to_fract (x * y) = map_poly to_fract x * map_poly to_fract y"
ct_prefixE: fixes xs :: "ct list"   and ys :: "ct list" assumes "ct_prefix xs ys" obtains as :: "ct list" and zs :: "ct list" where "ys = as @ zs" and "ct_list_eq as xs"
q_Disj_def: fixes x :: "hf"   and y :: "hf" shows "q_Disj x y \<equiv> \<langle>htuple 3, x, y\<rangle>"
isomorphic_reflexive: fixes f :: "'a" assumes "B.ide f" shows "B.isomorphic f f"
out_arcs_def: fixes G :: "('c, 'b) pre_digraph"   and v :: "'c" shows "out_arcs G v \<equiv> {e \<in> arcs G. tail G e = v}"
A: shows "S' (Inf X) = Inf (S' ` X)"
point_rep_number_alt_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and x :: "'b" assumes "incidence_system point_set block_collection" shows "block_collection rep x = int (size (filter_mset ((\<in>) x) block_collection))"
ct_suffix_imp_ct_list_eq: fixes xs :: "ct list"   and ys :: "ct list" assumes "ct_suffix xs ys" shows "ct_list_eq (drop (length ys - length xs) ys) xs"
bisimulation_flip: shows "bisimulation trsys2 trsys1 (flip bisim) (flip tlsim)"
some_in_set_if_set_neq_vempty: fixes A :: "V" assumes "A \<noteq> 0" shows "(SOME x. x \<in>\<^sub>\<circ> A) \<in>\<^sub>\<circ> A"
less_eq_funI: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "\<exists>c>0. \<exists>n. \<forall>m>n. f m \<le> c * g m" shows "f \<lesssim> g"
case1: shows "direction lineA = 0 \<longrightarrow> lineA \<parallel>  lineC"
eucl_of_list_map_mult_left: fixes xs :: "'a list"   and r :: "real"   and f :: "'a \<Rightarrow> real" assumes "length xs = DIM('b)" shows "eucl_of_list (map (\<lambda>x. r * f x) xs) = r *\<^sub>R eucl_of_list (map f xs)"
steps_def: fixes P :: "'a"   and Cal :: "'a processCalculus"   and Q :: "'a" shows "P \<longmapsto>Cal* Q \<equiv> \<exists>n. P \<longmapsto>Cal\<^bsup>n\<^esup> Q"
cmp_def: fixes \<mu>\<nu> :: "'b \<times> 'b" shows "cmp \<mu>\<nu> = (if B.VV.arr \<mu>\<nu> then G (F (fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)) \<cdot>\<^sub>D G (\<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D \<Phi>\<^sub>G (F (B.dom (fst \<mu>\<nu>)), F (B.dom (snd \<mu>\<nu>))) else D.null)"
list_graph_to_graph_def: fixes G :: "'c list_graph" shows "list_graph_to_graph G = \<lparr>nodes = set (nodesL G), edges = set (edgesL G)\<rparr>"
simps: fixes x :: "nat"   and s :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop x (s, l, r) = (if s = 0 then inv_loop0 x (l, r) else if s = 1 then inv_loop1 x (l, r) else if s = 2 then inv_loop2 x (l, r) else if s = 3 then inv_loop3 x (l, r) else if s = 4 then inv_loop4 x (l, r) else if s = 5 then inv_loop5 x (l, r) else if s = 6 then inv_loop6 x (l, r) else False)"