hcju/isabelleps huggingface dataset:
lm014: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G"   and "finite G" shows "\<forall>y\<in>Range a. finite y"
card_arcs_dv: shows "card (arcs (del_vert u)) = card (arcs G)"
nonneg_power_le: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.semiring_1_real_line one times plus zero minus less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq (power.power one times a n) (power.power one times b n) = (n = 0 \<or> less_eq a b)"
connected_components_of_empty_space: fixes X :: "'b topology" assumes "topspace X = {}" shows "connected_components_of X = {}"
wt_total_ground: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "ground t"   and "ground s" shows "wt t >\<^sub>p wt s \<or> wt s >\<^sub>p wt t \<or> wt t =\<^sub>p wt s"
presSwap_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresSwap h (fromMOD MOD)" shows "presSwap (fromIMor h) MOD"
pochhammer_minus: fixes b :: "'b"   and k :: "nat" shows "pochhammer (- b) k = (- (1::'b)) ^ k * pochhammer (b - of_nat k + (1::'b)) k"
purgeIdle_def: shows "GM_sec_model.purgeIdle \<equiv> filter isState"
gtrp_env_1: shows "c = add_mset s_ c_"
gso'_carrier_vec: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "gs.gso' i \<in> carrier_vec n"
divide_by_norm_row_code: fixes A :: "(('b, 'a) vec, 'c) vec"   and a :: "'c" shows "($) (divide_by_norm_row A a) = (\<lambda>b. ((1 / norm (column b A)) *\<^sub>R column b A) $ a)"
adjacent_singleton: fixes t :: "'b tree" assumes "t \<noteq> Leaf" shows "adjacent (l t, r t) [t]"
select_convs_1: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'b" shows "break_status \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = break_status"
is_ring_morphism: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f"   and U :: "'d set" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "is_open U" shows "ring_homomorphism (fam_morphisms U) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF>' U) (add_str' U) (mult_str' U) (zero_str' U) (one_str' U)"
embeddable_classI: fixes f :: "'a \<Rightarrow> V" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "OFCLASS('a, embeddable_class)"
type_universe_eq_iff: fixes t1 :: "pdf_type"   and t2 :: "pdf_type" shows "(type_universe t1 = type_universe t2) = (t1 = t2)"
matchT_ZO_def: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchT_ZO c d \<equiv> \<forall>s t s'. s \<approx> t \<and> (c, s) \<rightarrow>t s' \<longrightarrow> s' \<approx> t \<and> discr d \<or> (\<exists>d' t'. (d, t) \<rightarrow>c (d', t') \<and> s' \<approx> t' \<and> discr d') \<or> (\<exists>t'. (d, t) \<rightarrow>t t' \<and> s' \<approx> t')"
QuoteP_graph_aux_def: shows "QuoteP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>QuoteP_aux s t u k. x1 = (t, u) \<and> x2 = SyntaxN.Ex s (SyntaxN.Ex k (SeqQuoteP t u (Var s) (Var k))) \<and> atom s \<sharp> (t, u, k) \<and> atom k \<sharp> (t, u))"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'c list" assumes "i < |xs|" shows "take (Suc i) xs = take i xs @ [xs ! i]"
conjugate_square_ge_0_vec: fixes v :: "'a vec" shows "(0::'a) \<le> inner_prod v v"
lookup_update_with: fixes as :: "'d list"   and f :: "'c list \<Rightarrow> 'c list"   and t :: "('d, 'c list) trie"   and bs :: "'d list" shows "lookup_tries (update_with_tries as f t) bs = (if as = bs then f (lookup_tries t as) else lookup_tries t bs)"
eventually_lipschitz: fixes x0 :: "'a"   and c :: "real" assumes "t0 \<in> T"   and "x0 \<in> X"   and "0 < c" obtains L :: "real" where "\<forall>\<^sub>F u in at_right 0. \<forall>t'\<in>cball t0 (c * u) \<inter> T. L-lipschitz_on (cball x0 u \<inter> X) (f t')"
q1_sat_props: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p (take (length qs div 2) qs) (get_subsets_R (calculate_data_R p (take (length qs div 2) qs))) (get_signs_R (calculate_data_R p (take (length qs div 2) qs))) (get_matrix_R (calculate_data_R p (take (length qs div 2) qs)))"
drop_lift: fixes v :: "'d" shows "\<lceil>\<lfloor>v\<rfloor>\<rceil> = v"
normalize_mult_normalize_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize (times (normalize a) b) = normalize (times a b)"
red_image_monic: fixes B :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "punit.red (punit.monic ` B) = punit.red B"
P1_ideal_ddh1_\<sigma>_false: fixes \<sigma> :: "bool"   and M :: "'a \<times> 'a"   and z :: "'f"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)"   and D :: "'e \<times> 'a \<Rightarrow> bool spmf" assumes "\<sigma> = False" shows "P1_ideal_model M \<sigma> z \<A> \<bind> D = local.ddh.DDH1 (P1_DDH_mal_adv_\<sigma>_false M z \<A> D)"
fst_sup: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "fst (sup x y) = sup (fst x) (fst y)"
assms_3: fixes x :: "'b" assumes "x \<in> A" shows "f x dvd y"
PDQ_HMA: shows "(P_HMA, D_HMA, Q_HMA) = diagonalize_2x2 (Mod_Type_Connect.to_hma\<^sub>m A)"
GSubspace_is_FGModule: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and U :: "'c set" assumes "FGModule G smult V"   and "RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V" shows "FGModule G smult U"
Leaf_def: shows "Old_Datatype.Leaf \<equiv> Old_Datatype.Atom \<circ> Inl"
map_upt_len_conv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map (\<lambda>i. f (xs ! i)) [0..<length xs] = map f xs"
Compl_eq_Compl_iff: fixes A :: "'f set"   and B :: "'f set" shows "(- A = - B) = (A = B)"
less: fixes w :: "real" assumes "w \<in> {0..1}" shows "(1 - x * y) * w < 1"
red_lcl_incr: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P,E \<turnstile> \<langle>e,(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow> \<langle>e',(h\<^sub>1, l\<^sub>1)\<rangle>" shows "dom l\<^sub>0 \<subseteq> dom l\<^sub>1"
f_scale: fixes x :: "'a"   and c :: "complex" assumes "x \<in> S" shows "f (c *\<^sub>C x) = c *\<^sub>C f x"
not_strongly_preferred_iff: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "total_preorder_on carrier le"   and "a \<in> carrier"   and "b \<in> carrier" shows "(\<not> a \<prec>[le] b) = le b a"
eval_cnf_def: fixes \<Phi> :: "nat \<Rightarrow> bool"   and F :: "(nat \<times> bool) set set" shows "\<Phi> \<turnstile> F = (\<forall>C\<in>F. \<exists>(n, b)\<in>C. \<Phi> n = b)"
I_def: shows "I \<equiv> \<lambda>s. \<up> (0 \<le> s ''l'' \<and> (\<exists>k. s ''r'' - s ''l'' = 2 ^ k)) + enat (5 * Discrete.log (nat (s ''r'') - nat (s ''l'')))"
cone_add_cone: fixes C :: "'b vec set"   and n :: "nat" assumes "C \<subseteq> carrier_vec n" shows "gram_schmidt.cone n C + gram_schmidt.cone n C = gram_schmidt.cone n C"
simps_1: fixes uu :: "'g \<Rightarrow> bool" shows "find uu [] = None"
l3_inv2_trans: shows "{l3_inv2 \<inter> l3_inv1} TS.trans l3 {> l3_inv2}"
p_strict_mono: fixes x :: "real"   and y :: "real" assumes "x < y"   and "y \<le> delayed_safe_distance.other.t_stop" shows "delayed_safe_distance.other.p x < delayed_safe_distance.other.p y"
induct: fixes x :: "'e list"   and T :: "('e, 'f) TS_scheme"   and P :: "'e list \<Rightarrow> bool" assumes "x \<in> beh T"   and "P []"   and "\<And>s. s \<in> init T \<Longrightarrow> P [s]"   and "\<And>s b t. \<lbrakk>s # b \<in> beh T; P (s # b); (s, t) \<in> TS.trans T\<rbrakk> \<Longrightarrow> P (t # s # b)" shows "P x"
Elem_affine_of_ivl_le: fixes e :: "nat \<Rightarrow> real"   and l :: "'a"   and u :: "'a" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "l \<le> u" shows "l \<le> aform_val e (aform_of_ivl l u)"
invar_NodeR2: fixes l :: "('b \<times> nat) tree"   and x :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree"   and r' :: "('b \<times> nat) tree" assumes "invar \<langle>l, (x, n), r\<rangle>"   and "sngl r'"   and "n = lvl r + 1"   and "invar r'"   and "lvl r' = n" shows "invar \<langle>l, (x, n), r'\<rangle>"
m_le_z: fixes x :: "int" shows "- \<infinity> \<le> ant x"
emeasure_PiM_emb: fixes J :: "nat set"   and X :: "nat \<Rightarrow> 'a set" assumes "J \<subseteq> UNIV"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> M.events" shows "emeasure (Pi\<^sub>M UNIV (\<lambda>i. M)) (emb UNIV J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. emeasure M (X i))"
enum_less: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s"   and "i < n" shows "(kuhn_simplex.enum base upd i < a) = (kuhn_simplex.enum base upd (Suc i) \<le> a)"
assms_4: shows "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"
line_extension_mem_iff: fixes R :: "('a, 'b) ring_scheme"   and u :: "'a"   and K :: "'a set"   and a :: "'a"   and E :: "'a set" assumes "ring R" shows "(u \<in> ring.line_extension R K a E) = (\<exists>k\<in>K. \<exists>v\<in>E. u = k \<otimes>\<^bsub>R\<^esub> a \<oplus>\<^bsub>R\<^esub> v)"
that: fixes w\<^sub>2 :: "'item stream" assumes "w =\<^sub>I w\<^sub>1 @- w\<^sub>2" shows "thesis"
is_arg_max_linorder: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "is_arg_max f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> f y \<le> f x))"
null_sets_density_iff: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and A :: "'a set" assumes "f \<in> borel_measurable M" shows "(A \<in> null_sets (density M f)) = (A \<in> sets M \<and> (AE x\<in>A in M. f x = 0))"
assms_1: shows "a' = \<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>"
invar_fold_insert_nth: fixes pxs :: "(nat \<times> 'b) list"   and p :: "nat"   and xs :: "'b list"   and b :: "'b" assumes "\<forall>x\<in>set pxs. p < fst x"   and "p < length xs"   and "xs ! p = b" shows "fold (\<lambda>(x, y). insert_nth x y) pxs xs ! p = b"
induct: fixes P :: "int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>uu_. P uu_ []"   and "\<And>uv_ p. P uv_ [p]"   and "\<And>p p\<^sub>0 v va. P p (v # va) \<Longrightarrow> P p (p\<^sub>0 # v # va)" shows "P a0 a1"
inj_sum_decode: fixes A :: "nat set" shows "inj_on sum_decode A"
mat_add_comm: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<oplus> g = g \<oplus> f"
Nil_elem_T: fixes P :: "'b process" shows "[] \<in> T P"
simps_2: fixes m :: "'a \<Rightarrow> 'b set"   and ms :: "('a \<Rightarrow> 'b set) list" shows "\<Union>.m # ms = m \<union>. \<Union>.ms"
Schema_Sim: fixes sc1 :: "'d \<times> 'd \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'd"   and sc2 :: "'c \<times> 'c \<Rightarrow> bool" assumes "\<And>a. Pre sc1 (S a) \<Longrightarrow> Pre sc2 a"   and "\<And>a b. \<lbrakk>Pre sc1 (S a); sc2 (a, b)\<rbrakk> \<Longrightarrow> sc1 (S a, S b)" shows "Schema sc1 \<preceq>S Schema sc2"
degen_skip_arbitrary: fixes k :: "nat"   and cs :: "('b \<Rightarrow> bool) list"   and l :: "nat"   and w :: "'b stream" assumes "k < length cs"   and "l < length cs"   and "infs (degen cs) (w ||| k ## sscan (count cs) w k)" obtains u :: "'b list" and v :: "'b stream" where "w = u @- v" and "fold (count cs) u k = l"
diag_mat_def: fixes A :: "'a mat" shows "diag_mat A = map (\<lambda>i. A $$ (i, i)) [0..<dim_row A]"
zhmsetmset_le: fixes x :: "zhmultiset"   and xa :: "zhmultiset" shows "(zhmsetmset x \<le> zhmsetmset xa) = (x \<le> xa)"
st: shows "bezout_coefficients y (prod_list ys) = (s, t)"
meet_bot: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> \<bottom>\<^bsub>L\<^esub> = \<bottom>\<^bsub>L\<^esub>"
subsequence_def: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" shows "w [i \<rightarrow> j] \<equiv> map w [i..<j]"
join_eq: fixes n :: "nat"   and R :: "nat set"   and t1 :: "'a option list set"   and t2 :: "'a option list set"   and pos :: "bool" assumes "table n R t1"   and "table n R t2" shows "join t2 pos t1 = (if pos then t2 \<inter> t1 else t2 - t1)"
cexpr_typing_varE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c CVar x : t" obtains "t = \<Gamma> x"
states_subset: shows "\<delta>_states \<delta> \<subseteq> Q"
Pow_def: fixes A :: "'d set" shows "Pow A = {B. B \<subseteq> A}"
elims: fixes x :: "nat"   and xa :: "real mpoly"   and y :: "real mpoly \<times> real mpoly \<times> real mpoly" assumes "get_coeffs x xa = y" obtains var :: "nat" and xb :: "real mpoly" where "x = var" and "xa = xb" and "y = (isolate_variable_sparse xb var 2, isolate_variable_sparse xb var 1, isolate_variable_sparse xb var 0)"
assms: shows "functor (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F"
g_def: shows "g \<equiv> the_apartment_iso A' A''"
sorted_list_lemma: fixes a :: "real"   and b :: "real"   and n :: "nat"   and l :: "real list" assumes "a < b"   and "n + 1 < length l"   and "strict_sorted l"   and "l ! n < a"   and "b < l ! (n + 1)" shows "\<nexists>x. List.member l x \<and> a \<le> x \<and> x \<le> b"
append_assoc: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "append (append xs ys) zs = append xs (append ys zs)"
sumUnionDisjoint2: fixes X :: "'d set set"   and f :: "'d \<Rightarrow> 'e" assumes "\<forall>x\<in>X. finite x"   and "is_non_overlapping X" shows "sum f (\<Union> X) = sum (sum f) X"
preserves_modes_mem_def2: fixes \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set" shows "preserves_modes_mem \<R> = (\<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C \<and> mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C)"
rinf_uset: fixes p :: "fm"   and x :: "real"   and bs :: "real list" assumes "isrlfm p"   and "\<not> Ifm (x # bs) (minusinf p)"   and "\<not> Ifm (x # bs) (plusinf p)"   and "\<exists>x. Ifm (x # bs) p" shows "\<exists>(l, n)\<in>set (uset p). \<exists>(s, m)\<in>set (uset p). Ifm ((Inum (x # bs) l / real_of_int n + Inum (x # bs) s / real_of_int m) / 2 # bs) p"
raise_trap_privilege: fixes s :: "'b sparc_state"   and s' :: "'b sparc_state"   and t :: "Trap" assumes "get_S (cpu_reg_val PSR s) = 0 \<and> s' = snd (fst (raise_trap t s))" shows "get_S (cpu_reg_val PSR s') = 0"
enat_unfold_eq_0: fixes stop :: "'c \<Rightarrow> bool"   and "next" :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "(enat_unfold stop next a = 0) = stop a"
inorder_delete: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (delete x t) = del_list x (Tree2.inorder t)"
A: shows "\<forall>n<length xs. P (xs ! n) (drop (Suc n) xs)"
alpha_fresh': fixes a :: "name"   and x :: "'a"   and b :: "name"   and y :: "'a"   and c :: "name" assumes "[a].x = [b].y"   and "c \<noteq> a"   and "c \<noteq> b"   and "c \<sharp> x"   and "c \<sharp> y" shows "x = [(a, c)] \<bullet> [(b, c)] \<bullet> y"
TID_self_zero: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "Turn_into_distance f" shows "turn_into_distance f x x = 0"
k_ins_def: fixes ec :: "'rep"   and A :: "('ma, 'mt) AlgState" shows "k_ins ec A \<equiv> \<lparr>aActs = actsUpdate ec A, aTrans = foldr (transUpdate ec) (k_succs ec) (aTrans A)\<rparr>"
d_sq_norm_comparison: fixes n :: "nat"   and fs :: "int vec list"   and \<alpha> :: "rat"   and num :: "int"   and denom :: "int"   and i :: "nat" assumes "fs_int_indpt n fs"   and "quotient_of \<alpha> = (num, denom)"   and "i < length fs"   and "i \<noteq> 0" shows "(fs_int.d n fs i * fs_int.d n fs i * denom \<le> num * fs_int.d n fs (i - 1) * fs_int.d n fs (Suc i)) = (\<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2)"
continuous_on_floatarith: fixes fa :: "floatarith"   and xs :: "nat list"   and vs :: "real list" assumes "continuous_on_floatarith fa"   and "length xs = DIM('a)"   and "distinct xs" shows "continuous_on UNIV (\<lambda>x. interpret_floatarith fa (list_updates xs (list_of_eucl x) vs))"
i_append_nth: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "(xs \<frown> f) n = (if n < length xs then xs ! n else f (n - length xs))"
path_inv_Branching_Left: fixes p :: "bool list"   and l :: "tree"   and r :: "tree" shows "path (Left # p) (Branching l r) = path p l"
ide_map\<^sub>0_obj: fixes a :: "'a" assumes "C.obj a" shows "D.ide (F.map\<^sub>0 a)"
neq_lex_trans: fixes a :: "real \<times> real"   and b :: "real \<times> real" assumes "a \<noteq> b"   and "Counterclockwise_2D_Arbitrary.lex a b" shows "lexs a b"
rotate_conj_pow: fixes u :: "'b list"   and v :: "'b list"   and k :: "nat" shows "rotate \<^bold>|u\<^bold>| ((u \<cdot> v) \<^sup>@ k) = (v \<cdot> u) \<^sup>@ k"
diff_diff_add: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a b) c = minus a (plus b c)"
intros_1: fixes a :: "'a"   and r :: "('a \<times> 'a) set" shows "(a, a) \<in> r\<^sup>*"
cases: fixes x :: "v list" obtains env :: "v sem_env" and n :: "char list" and e :: "exp0" and v2 :: "v" where "x = [Closure env__ n__ e, v2__]"   | env :: "v sem_env" and funs :: "(char list \<times> char list \<times> exp0) list" and n :: "char list" and v2 :: "v" where "x = [Recclosure env funs n, v2]"   | "x = []"   | vb :: "lit" and va :: "v list" where "x = Litv vb__ # va__"   | vb :: "(char list \<times> tid_or_exn) option" and vc :: "v list" and va :: "v list" where "x = Conv vba__ vc__ # vaa__"   | vb :: "v sem_env" and vc :: "(char list \<times> char list \<times> exp0) list" and vd :: "char list" where "x = [Recclosure vbb__ vca__ vd__]"   | vb :: "v sem_env" and vc :: "(char list \<times> char list \<times> exp0) list" and vd :: "char list" and v :: "v" and vf :: "v" and vg :: "v list" where "x = Recclosure vbc__ vc vda__ # v__ # vf # vg"   | vb :: "nat" and va :: "v list" where "x = Loc vbd__ # vab__"   | vb :: "v list" and va :: "v list" where "x = Vectorv vbe__ # va"   | v :: "v" where "x = [ve__]"   | v :: "v" and vb :: "v" and vd :: "v" and ve :: "v list" where "x = v # vb # vd # ve"
yield: fixes s :: "'a"   and a :: "'b"   and s' :: "'a" assumes "g s = Yield a s'" shows "(s', s) \<in> R"
induct: fixes x1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and x2 :: "'d"   and x3 :: "'e \<Rightarrow> 'f"   and x4 :: "'d"   and x5 :: "'e \<Rightarrow> 'f"   and P :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint) \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow> \<langle>x4,x5\<rangle>"   and "\<And>A l u a l' u'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsub>a\<^esub> \<langle>l', u'\<rangle> \<Longrightarrow> P A l u l' u'"   and "\<And>A l u d l' u'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsup>d\<^esup> \<langle>l', u'\<rangle> \<Longrightarrow> P A l u l' u'" shows "P x1 x2 x3 x4 x5"
in_rel_def: fixes R :: "('a \<times> 'b) set"   and x :: "'a"   and y :: "'b" shows "in_rel R x y \<equiv> (x, y) \<in> R"
NotI: fixes v :: "i"   and \<phi> :: "\<o>" assumes "\<not> [\<phi> in v]" shows "[\<^bold>\<not>\<phi> in v]"
sbelow_rel_def: fixes A :: "('c \<times> 'd) set" shows "\<langle>A\<rangle>sbelow_rel = \<langle>A\<rangle>sctn_rel O br sbelow_halfspace top"
KL_density: fixes b :: "real"   and f :: "'a \<Rightarrow> real" assumes "1 < b"   and "f \<in> borel_measurable (density M (\<lambda>x. ennreal (f x)))"   and "AE x in density M (\<lambda>x. ennreal (f x)). 0 \<le> f x" shows "KL_divergence b (density M (\<lambda>x. ennreal (f x))) (density (density M (\<lambda>x. ennreal (f x))) (\<lambda>x. ennreal (f x))) = LINT x|density M (\<lambda>x. ennreal (f x)). f x * log b (f x)"
sets_eq_bot: fixes M :: "'b measure" shows "(sets M = {{}}) = (M = bot)"
right_neutral: fixes a :: "'b" shows "a \<cdot> (1::'b) = a"
linear_scale_self: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'a" assumes "vector_space scale" shows "Vector_Spaces.linear scale scale (scale c)"
one_is_same: fixes H :: "'a set" assumes "subgroup H G" shows "\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> = \<one>"
g_not_zero: shows "g \<noteq> \<zero>\<^bsub>P\<^esub>"