hcju/isabelleps huggingface dataset:
Min_Ej_le: fixes j :: "nat"   and e :: "nat"   and x :: "nat list" assumes "j < n"   and "e \<in> Ej j x"   and "length x \<le> m" shows "Min (Ej j x) \<le> Max (set a)"
tfinite_induct: fixes xs :: "('g, 'h) tllist"   and P :: "('g, 'h) tllist \<Rightarrow> bool" assumes "tfinite xs"   and "\<And>y. P (TNil y)"   and "\<And>x xs. \<lbrakk>tfinite xs; P xs\<rbrakk> \<Longrightarrow> P (TCons x xs)" shows "P xs"
group_size_lt_v: fixes G :: "'a set" assumes "G \<in> \<G>" shows "int (card G) < \<v>"
wtL_grdOf: fixes c :: "('fsym, 'psym) lit list"   and l :: "('fsym, 'psym) lit"   and x :: "Preliminaries.var" assumes "c \<in> \<Phi>"   and "l \<in> set c"   and "x \<in> nv2L l"   and "\<not> infTp (tpOfV x)" shows "wtL (grdOf c l x)"
d_delta_lnexp_cf6: fixes x :: "real" shows "((\<lambda>x. ln (exp_cf6 x) - x) has_real_derivative diff_delta_lnexp_cf6 x) (at x)"
simps: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'c" shows "r\<^sup>+\<^sup>+ a1 a2 = ((\<exists>a b. a1 = a \<and> a2 = b \<and> r a b) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>+\<^sup>+ a b \<and> r b c))"
word_msb_sint: fixes w :: "'b word" shows "msb w = (sint w < 0)"
group_closurep_group_closure_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "ab_group_add.group_closurep zero minus S = (\<lambda>x. x \<in> ab_group_add.group_closure zero minus S)"
closed_orbit_def: fixes x :: "'a" shows "closed_orbit x = (\<exists>T\<in>existence_ivl0 x. T \<noteq> 0 \<and> flow0 x T = x)"
split_minD: fixes t :: "('b \<times> color) tree"   and x :: "'b"   and t' :: "('b \<times> color) tree" assumes "split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "x # Tree2.inorder t' = Tree2.inorder t"
process_prop_4_def: fixes P :: "('c list \<times> 'c set) set \<times> 'c list set" shows "process_prop_4 P \<equiv> \<forall>xs x X. (xs, X) \<in> fst P \<longrightarrow> (xs @ [x], {}) \<in> fst P \<or> (xs, insert x X) \<in> fst P"
zero_lt_\<omega>\<^sub>z: shows "0 < \<omega>\<^sub>z"
hcomp_simps_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (H \<nu> \<mu>)" shows "partial_magma.dom V (H \<nu> \<mu>) = H (partial_magma.dom V \<nu>) (partial_magma.dom V \<mu>)"
bot_fun_def: shows "\<bottom> \<equiv> \<lambda>x. \<bottom>"
matches_to_match_list_normalize: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> m a p = match_list \<gamma> (normalize_match m) a p"
measurable_bind_prob_space2: fixes f :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd measure"   and R :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M prob_algebra N"   and "(\<lambda>(x, y). g x y) \<in> M \<Otimes>\<^sub>M N \<rightarrow>\<^sub>M prob_algebra R" shows "(\<lambda>x. f x \<bind> g x) \<in> M \<rightarrow>\<^sub>M prob_algebra R"
better_LOr2_intro: fixes y :: "name"   and x1 :: "name"   and N1 :: "trm"   and x2 :: "name"   and N2 :: "trm"   and b :: "coname"   and a :: "coname"   and M :: "trm" assumes "y \<sharp> ([x1].N1, [x2].N2)"   and "b \<sharp> [a].M" shows "Cut <b>.OrR2 <a>.M b y.OrL x1.N1 x2.N2 y \<longrightarrow>\<^sub>l Cut <a>.M x2.N2"
wf_below_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "wf_below r A = wf (Restr r ((r\<^sup>*)\<inverse> `` A))"
not_step1_Nil: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "\<not> step1 r xs []"
content_ff_eq_dff_nonzero: fixes p :: "'b fract poly"   and x :: "'b fract" assumes "content_ff p =dff x"   and "x \<noteq> 0" shows "p \<noteq> 0"
K_2: shows "ennreal (1 - d) < emeasure M (Ogood \<inter> {x \<in> space M. \<forall>n\<in>{1..Nf 1}. \<forall>l\<in>{1..n}. - (real K * real l) < u n x - u (n - l) x})"
fine_\<delta>_expand: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and \<delta>' :: "real \<Rightarrow> real" assumes "fine \<delta> (a, b) D"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<delta> x \<le> \<delta>' x" shows "fine \<delta>' (a, b) D"
rbl_add_suc_carry_fold: assumes "length (rev (to_bl x)) = length (rev (to_bl y))" shows "\<forall>car. (if car then rbl_succ else id) (rbl_add (rev (to_bl x)) (rev (to_bl y))) = foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip (rev (to_bl x)) (rev (to_bl y))) (\<lambda>_. []) car"
nth_shiftr: fixes w :: "'c word"   and m :: "nat"   and n :: "nat" shows "(w >> m) !! n = w !! (n + m)"
power_not_zero: fixes n :: "nat" assumes "n < LENGTH('b)" shows "2 ^ n \<noteq> 0"
conn2_1: shows "set (map of_int_hom.vec_hom fs') \<subseteq> Rn"
g_set_xy_loc_axioms: shows "g_set_xy_loc hs_ops hs_ops"
supp_swap: fixes a :: "atom"   and b :: "atom" shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
lowerdim_embeddings: assumes "DIM('c) < DIM('d)" obtains f :: "'c \<times> real \<Rightarrow> 'd" and g :: "'d \<Rightarrow> 'c \<times> real" and j :: "'d" where "linear f" and "linear g" and "\<And>z. g (f z) = z" and "j \<in> Basis" and "\<And>x. f (x, 0) \<bullet> j = 0"
disjoint_implies_not_part: fixes x :: "i"   and y :: "i" assumes "\<not> O x y" shows "\<not> P x y"
map_add_le_mapI: fixes f :: "'c \<Rightarrow> 'd option"   and h :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "f \<subseteq>\<^sub>m h"   and "g \<subseteq>\<^sub>m h" shows "f ++ g \<subseteq>\<^sub>m h"
preflect_def: shows "preflect \<equiv> \<lambda>t0. (-) (2 *\<^sub>R t0)"
ring_hom_monom_term: fixes a :: "'a"   and c :: "'a"   and S :: "('e, 'f) ring_scheme"   and h :: "'a \<Rightarrow> 'e"   and n :: "nat" assumes "a \<in> carrier R"   and "c \<in> carrier R"   and "ring S"   and "h \<in> ring_hom R S" shows "h (a \<otimes> c [^] n) = h a \<otimes>\<^bsub>S\<^esub> h c [^]\<^bsub>S\<^esub> n"
Image_singleton_iff: fixes b :: "'g"   and r :: "('h \<times> 'g) set"   and a :: "'h" shows "(b \<in> r `` {a}) = ((a, b) \<in> r)"
f01: fixes z :: "complex" assumes "z \<in> ball 0 1 - {0}" shows "f z \<noteq> 0 \<and> f z \<noteq> 1"
fpr_same_suffix': fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and prefix :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list"   and suffix :: "(ctx \<times> cld) list"   and prefix' :: "(ctx \<times> cld) list" assumes "find_path_rec_f P ctx cl prefix = Some path"   and "path = prefix @ suffix" shows "find_path_rec_f P ctx cl prefix' = Some (prefix' @ suffix)"
intersection_of_def: fixes P :: "'d set set \<Rightarrow> bool"   and Q :: "'d set \<Rightarrow> bool" shows "P intersection_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Inter> \<U> = S"
base_out: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "n \<le> i" shows "base i = p"
elims_2: fixes x :: "'b option"   and xa :: "'b option" assumes "x \<le> xa" obtains xb :: "'b" and y :: "'b" where "x = Some xb" and "xa = Some y__" and "xb \<le> y__"   | y :: "'b option" where "x = None" and "xa = y"
BIT_inv_def: fixes s :: "('b list \<times> bool list \<times> 'b list) pmf"   and x :: "'b"   and i :: "'b list" shows "BIT_inv s x i \<equiv> s = type0 i x (hd (filter (\<lambda>y. y \<noteq> x) i))"
2: shows "rawpsubst (rawpsubst \<phi>2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubst (rawpsubst \<phi>2 (zip (map Var vs2) (map snd txs))) (zip (map fst txs) vs2)"
app_def: fixes f :: "hf"   and x :: "hf" shows "happ f x = (THE y. \<langle>x, y\<rangle> \<^bold>\<in> f)"
meets_atrans: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "p \<parallel> q"   and "q \<parallel> r" shows "\<not> p \<parallel> r"
FVe: fixes e1 :: "expr"   and bop :: "bop"   and e2 :: "expr" shows "fv (e1 \<guillemotleft>bop\<guillemotright> e2) = fv e1 \<union> fv e2"
distinct_66: fixes pi1' :: "pi"   and pi2' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "pi1' \<parallel> pi2' \<noteq> [name1\<frown>name2]pi"
ss_inA: shows "\<forall>x\<in>set ss. x \<in> A"
VWO_iff_Ord_le: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "((\<beta>, \<alpha>) \<in> VWO) = (\<beta> \<le> \<alpha>)"
cong1: fixes xf :: "nat \<Rightarrow> real" shows "(\<lambda>f. evaluate_net (insert_weights s m1 (\<lambda>i. f (i + a))) (take (length (input_sizes (insert_weights s m1 (\<lambda>i. xf (i + a))))) inputs) $ j) = (\<lambda>f. evaluate_net (insert_weights s m1 (\<lambda>i. f (i + a))) (take (length (input_sizes (insert_weights s m1 (\<lambda>i. f (i + a))))) inputs) $ j)"
assms: shows "(v, w) \<in> (rel_restrict E R)\<^sup>+"
eventually_at_top_not_equal: fixes c :: "'c" shows "\<forall>\<^sub>F x in at_top. x \<noteq> c"
Sup_insert_insert: fixes a :: "real"   and b :: "real"   and s :: "real set" shows "Sup (insert a (insert b s)) = Sup (insert (max a b) s)"
connected_local_const: fixes A :: "'a set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "connected A"   and "a \<in> A"   and "b \<in> A"   and "\<forall>a\<in>A. \<forall>\<^sub>F b in at a within A. f a = f b" shows "f a = f b"
tllist_all2D_llist_all2_llist_of_tllist: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys" shows "llist_all2 P (llist_of_tllist xs) (llist_of_tllist ys)"
intro: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset" assumes "\<And>C. S C \<subseteq># C"   and "\<And>L C. L \<in># S C \<Longrightarrow> is_neg L" shows "selection S"
intro: fixes n :: "nat"   and fs :: "'a vec list" assumes "gram_schmidt_fs_Rn n fs"   and "gram_schmidt_fs_lin_indpt_axioms n fs" shows "gram_schmidt_fs_lin_indpt n fs"
retraction_def: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "retraction g \<equiv> \<exists>f. ide (g \<cdot> f)"
comp_assoc: fixes h :: "'b"   and g :: "'b"   and f :: "'b" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
rep_eq: fixes x :: "'a" shows "pmf (J x) = (\<lambda>y. R x y / escape_rate x)"
DD_def: fixes ars :: "('c \<times> 'd \<times> 'c) set"   and r :: "('d \<times> 'd) set"   and d :: "('c \<times> ('d \<times> 'c) list) \<times> ('c \<times> ('d \<times> 'c) list) \<times> ('c \<times> ('d \<times> 'c) list) \<times> 'c \<times> ('d \<times> 'c) list" shows "DD ars r d = (diagram ars d \<and> D2 r d)"
get_root_node_parent_child_rel: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_root_node child \<rightarrow>\<^sub>r root"   and "known_ptrs h"   and "type_wf h" shows "(root, child) \<in> (parent_child_rel h)\<^sup>*"
rel_distinct_2: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and y21 :: "'f"   and y22 :: "'h"   and x1 :: "'e" shows "\<not> rel_generat R1 R2 R3 (IO y21 y22) (Pure x1)"
stk_convert: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and ST :: "ty list"   and ST' :: "ty list" shows "P \<turnstile> ST [\<le>] ST' = Listn.le (subtype P) ST ST'"
hyps: fixes xs :: "'a list" shows "drop n (drop m_ xs) = drop (n + m_) xs"
record_call_def: fixes x :: "'b" shows "record_call x = (\<Lambda> ce. (\<lambda>y. if x = y then two_pred\<cdot>(ce y) else ce y))"
int_simps_5: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> \<not> P = P = #False"
hIm_hcomplex_of_complex: fixes z :: "complex" shows "hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)"
rb_aux_inv1_E: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and i :: "nat" assumes "rb_aux_inv1 bs"   and "i < length bs" obtains j :: "nat" where "j < length fs" and "lt (bs ! i) = lt (monomial (1::'b) (term_of_pair (0::'a, j)))" and "punit.lt (rep_list (bs ! i)) \<preceq> punit.lt (rep_list (monomial (1::'b) (term_of_pair (0::'a, j))))"   | p :: "'t \<Rightarrow>\<^sub>0 'b" and q :: "'t \<Rightarrow>\<^sub>0 'b" where "p \<in> set bs" and "q \<in> set bs" and "is_regular_spair p q" and "rep_list (spair p q) \<noteq> 0" and "lt (bs ! i) = lt (spair p q)" and "punit.lt (rep_list (bs ! i)) \<preceq> punit.lt (rep_list (spair p q))"
bisim: fixes s1 :: "'a"   and s2 :: "'b"   and x :: "'c" assumes "X s1 s2" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (oracle1 s1 x) (oracle2 s2 x)"
of_nat_le_0_iff: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) zero = (m = 0)"
assms_3: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. j \<le> k"   and "\<forall>j\<in>\<alpha> m - it. k \<le> j"   and "it \<subseteq> \<alpha> m"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
star_subid: fixes x :: "'a" assumes "x \<le> (1::'a)" shows "x\<^sup>\<star> = (1::'a)"
subset_Euclidean_space: fixes m :: "nat"   and n :: "nat" shows "(topspace (Euclidean_space m) \<subseteq> topspace (Euclidean_space n)) = (m \<le> n)"
zero_le_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
trimono_spec_defs_3: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "f \<circ> g = (\<lambda>x. f (g x))"
arg_min_least: fixes S :: "'b set"   and y :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "finite S"   and "S \<noteq> {}"   and "y \<in> S" shows "f (arg_min_on f S) \<le> f y"
freshAbs_swapAbs_id: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "goodAbs A"   and "freshAbs xs x1 A"   and "freshAbs xs x2 A" shows "A $[x1 \<and> x2]_xs = A"
ceq_uprod_def: shows "CEQ('b uprod) = map_option uprod_eq ceq_equal"
add_poly_p_add_mset_comb2: fixes A :: "(char list multiset \<times> int) multiset"   and Aa :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and ys :: "char list multiset"   and n :: "int"   and m :: "int" assumes "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#}) ({#}, {#}, r)" shows "add_poly_p\<^sup>*\<^sup>* (add_mset (ys, n) A, add_mset (ys, m) Aa, {#}) ({#}, {#}, add_mset (ys, n + m) r)"
simps: fixes f :: "char list \<Rightarrow> 'a"   and xs :: "char list list" shows "fun (FunDom f xs) = f"
mon_s_unconc: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and a :: "'b list"   and b :: "'b list" shows "mon_s fg (a @ b) = mon_s fg a \<union> mon_s fg b"
new_def: fixes p :: "('b, 'c) fm"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list" shows "new p a branch \<equiv> \<not> p at a in' set branch"
spmf_bind_leI: fixes p :: "'a spmf"   and f :: "'a \<Rightarrow> 'b spmf"   and x :: "'b"   and r :: "real" assumes "\<And>y. y \<in> set_spmf p \<Longrightarrow> spmf (f y) x \<le> r"   and "0 \<le> r" shows "spmf (p \<bind> f) x \<le> r"
switch_elem_card_le: fixes a :: "'c"   and A :: "'c set"   and b :: "'c" assumes "a \<in> A" shows "card (A - {a} \<union> {b}) \<le> card A"
countably_inf_bij: assumes "infinite UNIV"   and "infinite UNIV" shows "\<exists>b_of_a. bij b_of_a"
p'_def: shows "p' = Abs_poly (\<lambda>i. if i = n then 1 else c ^ (n - i - 1) * coeff p i)"
oreceived_msg_ok: fixes i :: "nat" shows "optoy i \<Turnstile> (otherwith nos_inc {i} (orecvmsg msg_ok), other nos_inc {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<sigma>, l). l \<in> {PToy-:1..} \<longrightarrow> msg_ok \<sigma> (msg (\<sigma> i)))"
relabel_measure: fixes u :: "nat" assumes "relabel_precond f l u" shows "sum_heights_measure (relabel_effect f l u) < sum_heights_measure l"
simps: fixes args :: "Monitor.args"   and nt :: "nat"   and t :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'a option list set) queue"   and data_in :: "(nat \<times> 'a option list set) queue"   and tuple_in :: "('a option list, nat) mapping"   and tuple_since :: "('a option list, nat) mapping" shows "shift_end args nt (t, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) = (let I = args_ivl args; data_prev' = dropWhile_queue (\<lambda>(t, X). right I < enat (nt - t)) data_prev; (data_in, discard) = takedropWhile_queue (\<lambda>(t, X). right I < enat (nt - t)) data_in; tuple_in = fold (\<lambda>(t, X) tuple_in. Mapping.filter (filter_cond X tuple_in t) tuple_in) discard tuple_in in (t, gc, maskL, maskR, data_prev', data_in, tuple_in, tuple_since))"
has_derivative_local_max: fixes f :: "'a \<Rightarrow> real"   and f' :: "'a \<Rightarrow> real"   and x :: "'a" assumes "(f has_derivative f') (at x)"   and "\<forall>\<^sub>F y in at x. f y \<le> f x" shows "f' = (\<lambda>h. 0)"
Sup_least: fixes A :: "'b set"   and z :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "Sup A \<le> z"
mset_un_cases: fixes a :: "'f"   and A :: "'f multiset"   and B :: "'f multiset" assumes "a \<in># A + B" obtains "a \<in># A"   | "a \<in># B"
rbt_del_in_tree: fixes t :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c"   and x :: "'a" assumes "rbt_sorted t"   and "inv1 t"   and "inv2 t" shows "((k, v) \<in> set (entries (rbt_del x t))) = (False \<or> x \<noteq> k \<and> (k, v) \<in> set (entries t))"
map_comp0: fixes g1 :: "'g \<Rightarrow> 'e"   and f1 :: "'c \<Rightarrow> 'g"   and g2 :: "'h \<Rightarrow> 'f"   and f2 :: "'d \<Rightarrow> 'h" shows "map_coG0 (g1 \<circ> f1) (g2 \<circ> f2) = map_coG0 g1 g2 \<circ> map_coG0 f1 f2"
dfs_induct: fixes S :: "'b"   and xs :: "'a list"   and P :: "'b \<Rightarrow> 'a list \<Rightarrow> bool" assumes "invariant S"   and "list_all is_node xs"   and "\<And>S. invariant S \<Longrightarrow> P S []"   and "\<And>S x xs. \<lbrakk>invariant S; is_node x; list_all is_node xs; memb x S \<Longrightarrow> P S xs; \<not> memb x S \<Longrightarrow> P (ins x S) (succs x @ xs)\<rbrakk> \<Longrightarrow> P S (x # xs)" shows "P S xs"
prems_1: shows "pre_subdivFace' g f u v n (ov # ovs)"
simps_1: fixes a :: "'c local.term" shows "\<^bold>\<I> \<^bold>\<Down> a = \<^bold>\<l>\<^bold>[a\<^bold>]"
span_superset: fixes S :: "('t \<Rightarrow>\<^sub>0 'f) set" shows "S \<subseteq> pmdl S"
bangPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and Rel' :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto>\<guillemotleft>Rel'\<guillemotright> S"   and "Rel \<subseteq> Rel'"   and "eqvt Rel'" shows "!P \<leadsto>\<guillemotleft>bangRel Rel'\<guillemotright> !Q"
le_numeral_extra_3: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
\<pi>0': shows "(\<pi> @\<^bsup>k\<^esup> \<pi>l) 0 = \<pi>' 0"
append_Cons: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "append (Cons x xs) ys = Cons x (append xs ys)"
path_connected_translationI: fixes S :: "'a set"   and a :: "'a" assumes "path_connected S" shows "path_connected ((+) a ` S)"
lr_of_tran_prereqs: fixes rt :: "32 routing_rule list"   and fw :: "32 simple_rule list"   and ifs :: "char list list"   and oft :: "(of_match_field, of_action list) flow_entry_match list" assumes "valid_prefixes rt"   and "simple_fw_valid fw"   and "lr_of_tran rt fw ifs = Inr oft" shows "list_all (all_prerequisites \<circ> ofe_fields) oft"
mult_ge1_powers_3: fixes a :: "'b" assumes "(1::'b) \<le> a" shows "(1::'b) \<le> a * a * a * a"
similar'_step_adm: fixes s :: "Value \<Rightarrow> CValue \<Rightarrow> bool" assumes "adm (\<lambda>x. s (fst x) (snd x))" shows "adm (\<lambda>x. similar'_step s (fst x) (snd x))"
ffield_vsubset_VUnion2: fixes r :: "V" shows "\<F>\<^sub>\<bullet> r \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> r))"
to_g: fixes x :: "complex" assumes "x \<in> S" shows "\<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod (f n y) \<le> h n)"
weak_respection_of_barbs_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs (symcl Rel) CWB"
squarefree_imp_prod_prime_factors_eq: fixes x :: "'a" assumes "squarefree x" shows "\<Prod>(prime_factors x) = normalize x"
simps_1: fixes x :: "'b" shows "is_compound (Var x) = False"
foldr_append2: fixes f :: "'d \<Rightarrow> 'c list"   and xs :: "'d list"   and ys :: "'c list"   and zs :: "'c list" shows "foldr ((@) \<circ> f) xs (ys @ zs) = foldr ((@) \<circ> f) xs ys @ zs"
OBTAIN_nofail: fixes P :: "'b \<Rightarrow> bool" shows "nofail (OBTAIN P) = (\<exists>x. P x)"
Sup_enat_minus1: fixes m :: "enat"   and A :: "enat set" assumes "m \<noteq> \<infinity>" shows "\<Squnion> A - m = (\<Squnion>n\<in>A. n - m)"
plossless_parallel_converter: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I> :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and \<I>2 :: "('m, 'n) \<I>"   and conv2 :: "('m, 'n, 'k, 'l) converter" assumes "plossless_converter \<I>1 \<I> conv1"   and "plossless_converter \<I>2 \<I> conv2"   and "\<I>1, \<I> \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>2, \<I> \<turnstile>\<^sub>C conv2 \<surd>" shows "plossless_converter (\<I>1 \<oplus>\<^sub>\<I> \<I>2) \<I> (conv1 |\<^sub>\<propto> conv2)"
abs_summable_on_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "(f abs_summable_on A) = (g abs_summable_on B)"
bind_split_asm: fixes P :: "'a option \<Rightarrow> bool"   and m :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "P (Option.bind m f) = (\<not> (m = None \<and> \<not> P None \<or> (\<exists>x. m = Some x \<and> \<not> P (f x))))"
m_widen: fixes y :: "'av"   and x :: "'av" assumes "\<not> y \<le> x" shows "m (x \<nabla> y) < m x"
simps_3: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst (\<phi> and\<^sub>n \<psi>) m = subst \<phi> m and\<^sub>n subst \<psi> m"
is_singleton_def: fixes A :: "'e set" shows "is_singleton A = (\<exists>x. A = {x})"
lt_of_nat_term_order_alt: fixes to :: "'c nat_term_order" shows "lt_of_nat_term_order to = (\<lambda>u v. ko.lt (key_order_of_nat_term_order_inv to) v u)"
Nml_implies_Arr: fixes t :: "'a local.term" assumes "Nml t" shows "Arr t"
lt_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(cmp x y = LESS) = (cmp y x = GREATER)"
antisymp_ord_option: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "antisymp ord" shows "antisymp (ord_option ord)"