hcju/isabelleps huggingface dataset:
x: shows "h \<turnstile> get_ancestors_si x \<rightarrow>\<^sub>r a # ancestors"
preorder_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.preorder less_eq less \<equiv> (\<forall>x y. less x y = (less_eq x y \<and> \<not> less_eq y x)) \<and> (\<forall>x. less_eq x x) \<and> (\<forall>x y z. less_eq x y \<longrightarrow> less_eq y z \<longrightarrow> less_eq x z)"
pr_algo_measure: fixes fl' :: "(nat \<times> nat \<Rightarrow> 'capacity) \<times> (nat \<Rightarrow> nat)"   and fl :: "(nat \<times> nat \<Rightarrow> 'capacity) \<times> (nat \<Rightarrow> nat)" assumes "(fl', fl) \<in> pr_algo_rel" shows "pr_algo_measure fl' < pr_algo_measure fl"
division_segment_euclidean_size: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "times (division_segment a) (semiring_1.of_nat one plus zero (euclidean_size a)) = a"
finite_imp_inj_to_nat_seg: fixes A :: "'c set" assumes "finite A" shows "\<exists>f n. f ` A = {i. i < n} \<and> inj_on f A"
ent_wandI: fixes Q :: "assn"   and P :: "assn"   and R :: "assn" assumes "Q * P \<Longrightarrow>\<^sub>A R" shows "P \<Longrightarrow>\<^sub>A Q -* R"
genby_mono: fixes T :: "'b set"   and S :: "'b set" assumes "order.greater_eq S T" shows "order.greater_eq \<langle>S\<rangle> \<langle>T\<rangle>"
not_min_gallery_double_crosses_wall: fixes Cs :: "'a set list" assumes "gallery Cs"   and "\<not> min_gallery Cs"   and "{} \<notin> set (wall_crossings Cs)" shows "\<not> distinct (wall_crossings Cs)"
continuous_on_compact_surface_projection: fixes S :: "'a set"   and V :: "'a set"   and d :: "'a \<Rightarrow> real" assumes "compact S"   and "S \<subseteq> V - {0::'a}"   and "cone V"   and "\<And>x k. x \<in> V - {0::'a} \<Longrightarrow> (0 < k \<and> k *\<^sub>R x \<in> S) = (d x = k)" shows "continuous_on (V - {0::'a}) (\<lambda>x. d x *\<^sub>R x)"
name_list_avoiding: fixes xvec :: "name list"   and c :: "'d" obtains pi :: "(name \<times> name) list" where "(pi \<bullet> xvec) \<sharp>* c" and "distinctPerm pi" and "set pi \<subseteq> set xvec \<times> set (pi \<bullet> xvec)"
all_nstri_imp_mul_nstri: fixes ys :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list" assumes "\<forall>i<length ys. snd (f (xs ! i) (ys ! i))"   and "length xs = length ys" shows "snd (mul_ext f xs ys)"
is_open_from_ind_is_open: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and U :: "'a set" assumes "ind_topology X is_open S"   and "is_open S"   and "ind_topology.ind_is_open X is_open S U" shows "is_open U"
is_faithful: fixes f :: "'c"   and f' :: "'c" assumes "C.par f f'"   and "G f = G f'" shows "f = f'"
subst_trm_ty_fresh: fixes X :: "tyvrs"   and t :: "trm"   and T :: "ty"   and Y :: "tyvrs" assumes "X \<sharp> t"   and "X \<sharp> T" shows "X \<sharp> t[Y \<mapsto>\<^sub>\<tau> T]"
simps_1: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b" shows "pre_digraph.awalk_verts G u [] = [u]"
sr: shows "order (complex_of_real sr) (charpoly cA) = 1"
spec_refine: fixes P :: "'d upred"   and R :: "'d upred"   and Q :: "'d upred" assumes "Q \<sqsubseteq> (P \<and> R)" shows "(P \<Rightarrow> Q) \<sqsubseteq> R"
K_in_maximal: fixes p :: "'b fm"   and V :: "'b fm set" assumes "\<turnstile> p"   and "V \<in> mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>})"   and "maximal V (mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>}))" shows "p \<in> V"
Max_less_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 (linorder.Max less_eq A) x = (\<forall>a\<in>A. less a x)"
bind_simps_1: fixes f :: "'b \<Rightarrow> 'a list" shows "List.bind [] f = []"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree list \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree list" assumes "\<And>t. P t []"   and "\<And>t\<^sub>1 t\<^sub>2 ts. (\<not> rank t\<^sub>1 < rank t\<^sub>2 \<Longrightarrow> P (link t\<^sub>1 t\<^sub>2) ts) \<Longrightarrow> P t\<^sub>1 (t\<^sub>2 # ts)" shows "P a0 a1"
shows_list_gen_def: fixes elt :: "'c \<Rightarrow> char list \<Rightarrow> char list"   and e :: "char list"   and l :: "char list"   and s :: "char list"   and r :: "char list"   and xs :: "'c list" shows "shows_list_gen elt e l s r xs = (if xs = [] then shows_string e else shows_between (shows_string l) (shows_sep elt (shows_string s) xs) (shows_string r))"
assms_3: shows "set (xs @ [x]) = set (ys @ [x] @ zs)"
to_gbg_alt_2: fixes m :: "'b" shows "g_E (to_gbg_ext m) = E"
ladder_last_n_of_cut: fixes L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" assumes "L \<noteq> []" shows "ladder_last_n (ladder_cut L n) = n"
filter_empty_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
assms_2: fixes i :: "nat" shows "d (seq i) \<le> n"
length_euler_slp_le: shows "2 * D \<le> length euler_slp"
assms_3: fixes x :: "'n" assumes "x \<in> S" shows "f x \<le> g x"
FCD_def: fixes \<Gamma> :: "'b Gamma"   and \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "FCD \<Gamma> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<beta> @ [c, v] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>' \<delta>'. set \<delta>' \<subseteq> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub> \<and> \<beta> @ \<delta>' @ [v] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
assms_3: fixes j :: "nat" assumes "j \<in> set inds" shows "j < length ls"
wffp_isState_doo_purgeIdle: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "isState (hd sl)" shows "doo (local.getGMState (hd sl)) (map getGMUserCom (tl (purgeIdle sl))) = local.getGMState (last sl)"
Exit_Def_empty: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and V :: "'i" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"   and "V \<in> Def Exit" shows "False"
mopup_aft_erase_c_aft_erase_a: fixes n :: "nat"   and l :: "cell list"   and xs :: "cell list"   and lm :: "nat list"   and ires :: "cell list" assumes "mopup_aft_erase_c (2 * n + 4, l, Bk # xs) lm n ires" shows "mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, xs) lm n ires"
prems_1: shows "extractFrame (!P) = \<langle>A\<^sub>b\<^sub>P, \<Psi>\<^sub>b\<^sub>P\<rangle>"
\<epsilon>_finite: shows "\<epsilon> \<noteq> \<top>"
sep_add'_def: fixes x :: "'b"   and y :: "'b" shows "sep_add' x y \<equiv> if x ## y then the (x \<oplus> y) else undefined"
truncation_def: fixes p :: "'node"   and S :: "'node set" shows "truncation p S \<equiv> {p'. reachable_through p S p'}"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. fst (w, f) \<star> snd (w, f)) local.map (\<lambda>f. \<ll> f \<cdot> (\<phi> \<star> local.dom f))"
mono_SucI1: fixes X :: "nat \<Rightarrow> 'a" assumes "\<forall>n. X n \<le> X (Suc n)" shows "monoseq X"
case_1: fixes f1 :: "'c"   and f2 :: "'c" shows "(case assign of assign \<Rightarrow> f1 | check \<Rightarrow> f2) = f1"
simps_10: fixes vl :: "'a \<Rightarrow> 'b fundef option"   and opinl :: "'opinl"   and \<Sigma> :: "Unboxed.type option list" shows "Subx.sp_instr vl (Ubx.instr.IOpInl opinl) \<Sigma> = Subx.sp_gen_pop_push (replicate (\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> opinl)) None, None) \<Sigma>"
Image_singleton_iff: fixes b :: "'e"   and r :: "('f \<times> 'e) set"   and a :: "'f" shows "(b \<in> r `` {a}) = ((a, b) \<in> r)"
B: fixes x :: "'b" shows "norm (g x) \<le> B * norm x"
prc_compat: fixes prc :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and prl :: "'a \<Rightarrow> bool"   and f :: "'a"   and g :: "'a"   and s1 :: "bool"   and ns1 :: "bool"   and h :: "'a"   and s2 :: "bool"   and ns2 :: "bool"   and s :: "bool"   and ns :: "bool" assumes "precedence prc prl"   and "prc f g = (s1, ns1)"   and "prc g h = (s2, ns2)"   and "prc f h = (s, ns)" shows "(ns1 \<and> ns2 \<longrightarrow> ns) \<and> (ns1 \<and> s2 \<longrightarrow> s) \<and> (s1 \<and> ns2 \<longrightarrow> s)"
distinct_45: fixes x5 :: "nat"   and x6 :: "msg" shows "Key x5 \<noteq> Hash x6"
fmap_bind: fixes f :: "'g \<rightarrow> 'e"   and xs :: "'h\<cdot>'f"   and k :: "'h \<rightarrow> 'g\<cdot>'f" shows "fmap\<cdot>f\<cdot>(xs \<bind> k) = xs \<bind> (\<Lambda> x. fmap\<cdot>f\<cdot>(k\<cdot>x))"
Quotient3_int: shows "Quotient3 (\<approx>) abs_int rep_int"
step: fixes f' :: "'c"   and x :: "'b"   and y :: "'a" assumes "\<And>x. U f' x = U f' x"   and "\<not> P x y" shows "y \<in> U (F (C (sup (\<lambda>x. {y. \<not> P x y}) (U f)))) x"
chine_assoc_def: shows "\<mu>\<nu>_\<pi>_\<rho>.chine_assoc \<equiv> \<langle>\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1\<^sub>1 \<lbrakk>\<mu>\<nu>.leg0, \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1\<rbrakk> \<langle>\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 \<lbrakk>\<pi>.leg0, \<rho>.leg1\<rbrakk> \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0\<rangle>\<rangle>"
uint_of_uint64_def: shows "uint_of_uint64 \<equiv> map_fun Rep_uint64 Abs_uint ucast"
rev_is_Nil_conv: fixes xs :: "'b list" shows "(rev xs = \<epsilon>) = (xs = \<epsilon>)"
infinite: fixes A :: "'e set" assumes "infinite A" shows "mset_set A = {#}"
no_loose_bvar_no_incr: fixes t :: "Core.term"   and lev :: "nat"   and inc :: "nat" assumes "\<not> loose_bvar t lev" shows "incr_bv inc lev t = t"
\<I>_eqI: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('a, 'b) \<I>" assumes "outs_\<I> \<I> = outs_\<I> \<I>'"   and "\<And>x. x \<in> outs_\<I> \<I>' \<Longrightarrow> responses_\<I> \<I> x = responses_\<I> \<I>' x" shows "\<I> = \<I>'"
hyps: fixes cs :: "'edge list" shows "upd_rev_cs cs (a # as') = upd_rev_cs cs as'"
SUP_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'd"   and D :: "'b \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
invariantE: fixes P :: "'a \<Rightarrow> bool"   and succs :: "'a \<Rightarrow> 'a list"   and g' :: "'a"   and g :: "'a" assumes "invariant P succs"   and "g' \<in> set (succs g)"   and "P g" shows "P g'"
inf_sup_absorb: fixes 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 x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (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)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) x (y::'b::type)) = x"
nth_eq_iff_index_eq: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "(xs ! i = xs ! j) = (i = j)"
s: shows "tensor_P (1\<^sub>m N) M1 + tensor_P (1\<^sub>m N) M0 = 1\<^sub>m d"
no_head_generates0: fixes v :: "'a list"   and u :: "'a list"   and b :: "'a list" assumes "v \<in> \<langle>\<C>\<rangle>" shows "u \<noteq> \<epsilon> \<longrightarrow> u \<in> \<langle>no_head_gen \<C> b\<rangle> \<longrightarrow> u \<cdot> v \<in> \<langle>no_head_gen \<C> b\<rangle>"
circline_point_0_def: shows "circline_point_0 \<equiv> abs_circline circline_point_0_clmat"
rep_eq: fixes x :: "uint8" shows "Rep_uint32 (uint32_of_uint8 x) = ucast (Rep_uint8 x)"
in_new_spairsD_2: fixes a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "Inl (a, b) \<in> set (new_spairs bs p)" shows "b \<in> set bs"
cyclic_groupI0: fixes a :: "'a" assumes "a \<in> carrier G"   and "carrier G = generate G {a}" shows "Finite_And_Cyclic_Groups.cyclic_group G a"
M_def: shows "M \<equiv> \<lambda>l B. trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l B) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l B)"
vbrelation_vinE: fixes r :: "V"   and x :: "V" assumes "vbrelation r"   and "x \<in>\<^sub>\<circ> r" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
correct_convergE: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "correct ((f, xs, ls) # rest, None)"   and "eval f xs \<down>" shows "reachable ((f, xs, ls) # rest, None) (rest, eval f xs)"
Suc_times_binomial_add: fixes a :: "nat"   and b :: "nat" shows "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
LIMSEQ_le_const: fixes X :: "nat \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b" assumes "X \<longlonglongrightarrow> x"   and "\<exists>N. \<forall>n\<ge>N. a \<le> X n" shows "a \<le> x"
elims: fixes x :: "nat"   and y :: "nat list" assumes "Nat_Bijection.list_decode x = y" obtains "x = 0" and "y = []"   | n :: "nat" where "x = Suc n" and "y = (case prod_decode n of (x, y) \<Rightarrow> x # Nat_Bijection.list_decode y)"
notin: shows "per_union R x y `` {y} \<notin> (\<Union>x\<in>V - R `` {x} - R `` {y}. {R `` {x}})"
first_def: fixes x :: "('b, 'c) OrdSum" shows "first x = fst (Rep_OrdSum x)"
t_wise_balance_axioms: shows "t_wise_balance V B 2 \<Lambda>"
fin1: shows "finite (X \<rightarrow>\<^sub>E Roots)"
fbox_g_odei: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> bool"   and U :: "'a \<Rightarrow> real set"   and S :: "'a set"   and t\<^sub>0 :: "real"   and Q :: "'a \<Rightarrow> bool" assumes "P \<le> I"   and "I \<le> |x\<acute>=f & G on U S @ t\<^sub>0] I"   and "(\<lambda>s. I s \<and> G s) \<le> Q" shows "P \<le> |x\<acute>=f & G on U S @ t\<^sub>0 DINV I ] Q"
lookup_pair_eq_valueI: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "(k, v) \<in> set xs" shows "comparator.lookup_pair comp xs k = v"
1: shows "\<forall>e. ListMem e as \<longrightarrow> varset_action e (prob_dom PROB - vs)"
distinct_1: fixes x1 :: "'f"   and x21 :: "'d \<times> 'e"   and x22 :: "('d, 'e, 'f) ctree fset" shows "Link x1 \<noteq> cNode x21 x22"
bernoulli_fps_aux: shows "bernoulli_fps * (fps_exp (1::'a) - 1) = fps_X"
map_tailrec_def: fixes f :: "'h \<Rightarrow> 'g"   and as :: "'h list" shows "map_tailrec f as = rev (map_tailrec_rev f as [])"
dims_mat_tensorlist_mult: fixes T :: "real tensor"   and A :: "real mat"   and Ts :: "real tensor vec"   and ds :: "nat list" assumes "T \<in> set\<^sub>v (mat_tensorlist_mult A Ts ds)" shows "dims T = ds"
Ord_cases: fixes k :: "hf" assumes "Ord k" obtains "k = 0"   | l :: "hf" where "Ord l" and "succ l = k"
leq_prodE: fixes p :: "'c \<times> 'd"   and q :: "'c \<times> 'd" assumes "p \<sqsubseteq> q" obtains "fst p \<sqsubseteq> fst q" and "snd p \<sqsubseteq> snd q"
integrable_const_ivl: fixes c :: "'b"   and a :: "'a"   and b :: "'a" shows "(\<lambda>x. c) integrable_on {a..b}"
evaln_elim_cases_11: fixes G :: "prog"   and s :: "st"   and T :: "ty"   and e :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (Cast T e)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains va :: "val" and s1 :: "state" where "v = In1 va" and "s' = abupd (raise_if (\<not> G,snd s1\<turnstile>va fits T) ClassCast) s1" and "G\<turnstile>Norm s \<midarrow>e-\<succ>va\<midarrow>n\<rightarrow> s1"
intros_1: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame G" shows "ParityGame.path_conforms_with_strategy G p LNil \<sigma>"
rep_mult_line_rep_non_zero: fixes A :: "cltn2"   and l :: "proj2_line" shows "cltn2_rep A *v proj2_line_rep l \<noteq> 0"
inres_simps_3: fixes x :: "'a" shows "inres (RETURN x) = (=) x"
functor_is_transformation: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'd" assumes "functor A B F" shows "natural_transformation A B F F F"
completion_distr_eq: fixes X :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "X \<in> completion M \<rightarrow>\<^sub>M N"   and "null_sets (distr (completion M) N X) = null_sets N" shows "completion (distr (completion M) N X) = distr (completion M) (completion N) X"
le_cong: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x .=\<^bsub>L\<^esub> y"   and "z .=\<^bsub>L\<^esub> w"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L"   and "w \<in> carrier L" shows "(x \<sqsubseteq>\<^bsub>L\<^esub> z) = (y \<sqsubseteq>\<^bsub>L\<^esub> w)"
pderiv_add: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "p \<in> carrier (UP Zp)"   and "q \<in> carrier (UP Zp)" shows "pderiv (p \<oplus>\<^bsub>UP Zp\<^esub> q) = pderiv p \<oplus>\<^bsub>UP Zp\<^esub> pderiv q"
refl_not_SN: fixes x :: "'a"   and R :: "('a \<times> 'a) set" assumes "(x, x) \<in> R" shows "\<not> SN R"
LPDs_alt: fixes r :: "mregex" shows "LPDs r = lfp (\<lambda>S. insert r S \<union> (insert r S \<bind> LPD))"
gkern2: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "ginj\<^bsub>F,G\<^esub> f" shows "gker\<^bsub>F,G\<^esub> f = {\<one>\<^bsub>F\<^esub>}"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_right_move ires rs (l, r) = (\<exists>ln rn. l = Oc # ires \<and> r = Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
access_matrix_complete: fixes V :: "('a word \<times> 'a wordinterval) list"   and E :: "('a word \<times> 'a word) list"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and s :: "'a word"   and d :: "'a word" assumes "(V, E) = access_matrix c rs"   and "runFw s d c rs = Decision FinalAllow" shows "\<exists>s_repr d_repr s_range d_range. (s_repr, d_repr) \<in> set E \<and> map_of V s_repr = Some s_range \<and> s \<in> wordinterval_to_set s_range \<and> map_of V d_repr = Some d_range \<and> d \<in> wordinterval_to_set d_range"
fw_middle_id: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and j' :: "nat"   and i' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "i \<le> n"   and "j \<le> n"   and "j' < j"   and "i' \<le> i" shows "fw m n 0 i' j' i j = m i j"
utp_theory_unital_axioms: shows "utp_theory_unital \<H> \<I>\<I>"
pmf_mix_deeper: fixes a :: "real"   and p :: "'a pmf"   and q :: "'a pmf"   and x :: "'a" assumes "a \<in> {0..1}" shows "pmf (mix_pmf a p q) x = a * pmf p x + pmf q x - a * pmf q x"
assms_2: shows "callee_invariant_on callee1 I1 \<I>"
web_weight_update: fixes w :: "'v \<Rightarrow> ennreal" assumes "\<And>x. \<not> vertex \<Gamma> x \<Longrightarrow> w x = 0"   and "\<And>x. w x \<noteq> \<top>" shows "web (\<Gamma>\<lparr>weight := w\<rparr>)"
evaldjf_qf: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> fm" assumes "\<forall>x\<in>set xs. qfree (f x)" shows "qfree (evaldjf f xs)"
FDOM_eff_subset_prob_dom_pair: fixes a :: "('c, 'd) fmap \<times> ('c, 'e) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'e) fmap) set" assumes "a \<in> PROB" shows "fmdom' (snd a) \<subseteq> prob_dom PROB"
OclNot_inject: fixes x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "not x = not y" shows "x = y"
"0_3": shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) C"
assms_7: fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "(f has_derivative g' tx) (at tx)"
Chain_mem_fTo: fixes C :: "'a set" assumes "Chain D C" shows "C \<in> carrier fTo D"
upto_generator_def: shows "local.upto_generator \<equiv> Generator (\<lambda>n. n \<le> bound, \<lambda>n. (n, n + 1))"
expectation_pair_pmf_snd: fixes p :: "'b pmf"   and q :: "'c pmf"   and f :: "'c \<Rightarrow> 'a" shows "measure_pmf.expectation (pair_pmf p q) (\<lambda>x. f (snd x)) = measure_pmf.expectation q f"
assms_3: shows "sorted (map (drop (i + 1)) xss)"
TOTAL_subseteq_CONS: shows "TOTAL \<subseteq> CONS"
2: shows "card {z \<in> A. z < y} < card {z \<in> A. z < x}"
zimplode_def: fixes A :: "ZF zet" shows "zimplode A \<equiv> implode (Rep_zet A)"
is_env_def: fixes MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U" shows "is_env MI e = (\<forall>x. e x \<in> fst MI)"
is_nth_power_mult: fixes n :: "nat"   and a :: "'a"   and b :: "'a" assumes "is_nth_power n a"   and "is_nth_power n b" shows "is_nth_power n (a * b)"
llist_all2_lmap1: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'e"   and xs :: "'g llist"   and ys :: "'f llist" shows "llist_all2 P (lmap f xs) ys = llist_all2 (\<lambda>x. P (f x)) xs ys"
card_basis_left_null_space_eq_dim: fixes A :: "(('a, 'c) vec, 'b) vec" shows "card (basis_left_null_space A) = vec.dim (left_null_space A)"
IH: fixes s :: "'a" assumes "s \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! Suc pc"   and "s \<in> A"   and "s \<noteq> \<top>" shows "wtl ls c (Suc pc) s \<noteq> \<top>"
get_heap_returns_result: fixes h :: "'a"   and f :: "'a \<Rightarrow> ('a, 'b, 'c) prog"   and x :: "'c" shows "h \<turnstile> Heap_Error_Monad.bind get_heap f \<rightarrow>\<^sub>r x = h \<turnstile> f h \<rightarrow>\<^sub>r x"
Derives1: shows "Derives1 (u @ v) (fst d + length u) (snd d) (u @ x)"
is_lb_insert: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and A :: "'c set"   and y :: "'c" assumes "class.po below" shows "po.is_lb below (insert x A) y = (below y x \<and> po.is_lb below A y)"
simps_8: fixes f :: "'c \<Rightarrow> 'b" shows "map_tree f Leaf = Leaf"
ideal_subset: fixes I :: "'a set"   and h :: "'a" assumes "ideal R I"   and "h \<in> I" shows "h \<in> carrier R"
sym_reachableI_in_awalk: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w1 :: "'a"   and w2 :: "'a" assumes "awalk u p v"   and "w1 \<in> set (awalk_verts u p)"   and "w2 \<in> set (awalk_verts u p)" shows "w1 \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> w2"
is_span: shows "C.span \<mu>\<nu>_\<pi>.leg0 \<mu>\<nu>_\<pi>.leg1"