hcju/isabelleps huggingface dataset:
range_focusI: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "p \<in> P[X]"   and "lookup p ` keys p \<subseteq> P[- X]" shows "p \<in> range (focus X)"
simps_3: fixes a :: "'d"   and x :: "'c"   and s :: "'e" shows "fix_step a (Yield x s) = Yield x (a, s)"
inter_aform_plane_refine_unoverloaded0: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and d :: "nat"   and xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and si :: "real list sctn"   and s :: "real list sctn"   and optns :: "nat" assumes "class.enum enum enum_all enum_ex"   and "d = CARD('a)"   and "(xi, x) \<in> aforms_rel"   and "(si, s) \<in> \<langle>rl_rel\<rangle>sctn_rel"   and "length xi = d"   and "length (normal si) = d" shows "(nres_of (inter_aform_plane_lv (length xi) optns xi si), inter_sctn_specs d x s) \<in> \<langle>aforms_rel\<rangle>nres_rel"
bisim1FAccNull: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and xs :: "'addr val list" shows "P,e\<bullet>F{D},h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, length (compE2 e), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
block_designI: fixes k :: "nat" assumes "\<And>bl. bl \<in># arcs_blocks \<Longrightarrow> card bl = k" shows "block_design (verts G) arcs_blocks (int k)"
assms_2: shows "xs' \<equiv> filter (\<lambda>z. snd z \<noteq> 0) xs"
simps_2: fixes l1 :: "'a tree"   and x :: "'a"   and r1 :: "'a tree"   and t2 :: "'a tree" shows "mrbst_diff \<langle>l1, x, r1\<rangle> t2 = (case split_bst' x t2 of (sep, l2, r2) \<Rightarrow> mrbst_diff l1 l2 \<bind> (\<lambda>l. mrbst_diff r1 r2 \<bind> (\<lambda>r. if sep then mrbst_join l r else return_pmf \<langle>l, x, r\<rangle>)))"
push_bit_of_int: fixes n :: "nat"   and k :: "int" shows "push_bit n (of_int k) = of_int (push_bit n k)"
not_prfOf_PPf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" assumes "prf \<in> proof"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "\<not> prfOf prf \<phi>" shows "bprv (neg (PPf (encPf prf) \<langle>\<phi>\<rangle>))"
INF_image: fixes g :: "'d \<Rightarrow> 'b \<times> 'c"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "Inf_prod (g ` f ` A) = Inf_prod ((g \<circ> f) ` A)"
dioph_dfs_start: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "fst (dioph_dfs n ks l) ! int_encode l = Some 0"
assms_1: shows "(Pos x, Neg x) \<in> (imp_graph cnf)\<^sup>*"
eval_prod_lam: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>g : HF'.prod a b \<rightarrow> c\<guillemotright>" shows "local.comp (eval b c) (HF'.prod (\<Lambda> a b c g) b) = g"
prime_imp_nonzero: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "ASSUMPTION (normalization_semidom.prime zero times one normalize x)" shows "x \<noteq> zero"
evaluation_functor_axioms: shows "FreeMonoidalCategory.evaluation_functor C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"
nn_integral_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>N (restrict_space M \<Omega>) f = set_nn_integral M \<Omega> f"
C_eq_s_ext: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "p \<noteq> []" shows "Cp (list2FWpolicy (separate p)) a = Cp (list2FWpolicy p) a"
fmap_ex_cof: fixes f :: "'a \<Rightarrow> 'b option"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> 'b option \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>l\<in>dom f. \<exists>L. finite L \<and> (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p (f l) l)" shows "\<exists>L. finite L \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p (f l) l)"
coprime_self_gauss_cnj: fixes z :: "gauss_int" assumes "coprime (ReZ z) (ImZ z)"   and "odd (gauss_int_norm z)" shows "coprime z (gauss_cnj z)"
oexp_Limit_le: fixes \<beta> :: "V"   and \<gamma> :: "V"   and \<alpha> :: "V" assumes "\<beta> < \<gamma>"   and "Limit \<gamma>"   and "Ord \<beta>"   and "0 < \<alpha>" shows "\<alpha> \<up> \<beta> \<le> \<alpha> \<up> \<gamma>"
division_ofD_3: fixes s :: "'a set set"   and i :: "'a set"   and K :: "'a set" assumes "s division_of i"   and "K \<in> s" shows "K \<noteq> {}"
graph_union_iff: fixes G\<^sub>1 :: "('b, 'c) labeled_graph"   and G\<^sub>2 :: "('b, 'c) labeled_graph" shows "(graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2) = (edges G\<^sub>1 \<subseteq> edges G\<^sub>2 \<and> vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2)"
while_eval_elim': fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and e :: "'d"   and c :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars"   and "(((While e c, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = mds \<and> mem' = mem"
sum4sq_int_def: shows "sum4sq_int = (\<lambda>(a, b, c, d). a\<^sup>2 + b\<^sup>2 + c\<^sup>2 + d\<^sup>2)"
leq_add_right: fixes x :: "'a"   and y :: "'a" shows "x \<le> x + y"
simps_1: fixes n\<^sub>s :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "\<exists>n'<n\<^sub>s. \<not> Option.is_none (devBC t n nid n')"   and "Option.is_none (devBC t n nid n\<^sub>s)" shows "devExt t n nid n\<^sub>s 0 = bc (\<sigma>\<^bsub>the (devBC t n nid (GREATEST n'. n' < n\<^sub>s \<and> \<not> Option.is_none (devBC t n nid n')))\<^esub>t (GREATEST n'. n' < n\<^sub>s \<and> \<not> Option.is_none (devBC t n nid n')))"
assms_4: shows "\<langle>g ` I\<rangle>\<^bsub>G\<^esub> = carrier G"
D: shows "(xs @ y # ys, Y) \<in> con_comp_failures P Q p q"
add_less_same_cancel1: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b + a \<subset># b) = (a \<subset># {#})"
"4_1": assumes "comp kx ky = Eq" shows "prod_ord_pair P xs ys = (lex_ord_pair (\<lambda>k x y. if P k x y then Some Eq else None) xs ys = Some Eq)"
trace_def: fixes a :: "'b"   and f :: "'b perm" shows "trace a f = map (\<lambda>n. (f ^ n) \<langle>$\<rangle> a) [0..<order f a]"
minus_diff_eq: fixes a :: "'a"   and b :: "'a" shows "- (a - b) = b - a"
intro: fixes hom :: "'b \<Rightarrow> 'a" assumes "hom (1::'b) = (1::'a)" shows "one_hom hom"
homology_homotopy_axiom: fixes S :: "'b set"   and T :: "'c set"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and p :: "int" assumes "homotopic_with (\<lambda>h. h ` S \<subseteq> T) X Y f g" shows "hom_induced p X S Y T f = hom_induced p X S Y T g"
added_ids_Deliver_Add_same_collapse: fixes i :: "'d"   and j :: "'d"   and e :: "'b" shows "added_ids [Deliver (i, Add j e)] e = [j]"
pairs_bot_closed: shows "stone_phi.pairs_bot \<in> stone_phi.pairs"
stake_snth: fixes n :: "nat"   and w :: "'a stream"   and x :: "'a stream"   and i :: "nat" assumes "stake n w = stake n x"   and "Suc i \<le> n" shows "w !! i = x !! i"
unR_def: fixes list_R1 :: "'b list_R1" shows "unR list_R1 = (case list_R1 of list_R1 x \<Rightarrow> x)"
bit_0: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a 0 = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
swap_cols_rows_index: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and a :: "nat"   and b :: "nat" assumes "i < dim_row A"   and "i < dim_col A"   and "j < dim_row A"   and "j < dim_col A"   and "a < dim_row A"   and "b < dim_row A" shows "swap_cols_rows a b A $$ (i, j) = A $$ (if i = a then b else if i = b then a else i, if j = a then b else if j = b then a else j)"
mpoly_to_poly_inverse: fixes p :: "'b mpoly"   and v :: "nat" assumes "vars p \<subseteq> {v}" shows "poly_to_mpoly v (mpoly_to_poly v p) = p"
simps_5: fixes v :: "nat" shows "bs_observe (BNat v) OFun = False"
Sigma_msetE: fixes c :: "'a \<times> 'b"   and A :: "'a multiset"   and B :: "'a \<Rightarrow> 'b multiset" assumes "c \<in># Sigma_mset A B" obtains x :: "'a" and y :: "'b" where "x \<in># A" and "y \<in># B x" and "c = (x, y)"
mult_comp_portfolio: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" assumes "portfolio pf1" shows "portfolio (qty_mult_comp pf1 qty)"
has_integral_0_closure_imp_0: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "continuous_on (closure S) f"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"   and "0 < emeasure lborel S"   and "emeasure lborel S < \<infinity>"   and "emeasure lborel (closure S) = emeasure lborel S"   and "open S"   and "(f has_integral 0) (closure S)"   and "x \<in> closure S" shows "f x = 0"
rtranclE: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> r\<^sup>*" obtains "a = b"   | y :: "'b" where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
ediv_cancel3: fixes k :: "nat"   and A :: "enat" assumes "0 < k" shows "A \<le> enat k * ediv A k"
leftderives_padfront: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and u :: "symbol list" assumes "leftderives \<alpha> \<beta>"   and "is_word u" shows "leftderives (u @ \<alpha>) (u @ \<beta>)"
dim_eucl_tangent_space: fixes a :: "'c" shows "dim (manifold_eucl.tangent_space \<infinity> a) = DIM('c)"
simps_4: fixes e\<^sub>1 :: "char list exp"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" shows "fv (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
ih: shows "finprod G f F [^] k = finprod G ((\<lambda>a. a [^] k) \<circ> f) F"
length_flip: fixes xs :: "bool list"   and i :: "nat" shows "length (flip xs i) = length xs"
extremum_strict: fixes a :: "'b" shows "\<not> order.greater bot a"
MemI: fixes i :: "name"   and j :: "name" shows "ss_fm (Var i IN Var j)"
weak_unit_cancel_right: fixes a :: "'a"   and f :: "'a"   and g :: "'a" assumes "weak_unit a"   and "ide f"   and "ide g"   and "f \<star> a \<cong> g \<star> a" shows "f \<cong> g"
YX: shows "simple_distributed M (\<lambda>x. (Y x, X x)) (\<lambda>x. prob ((\<lambda>x. (Y x, X x)) -` {x} \<inter> space M))"
CC_def: fixes E' :: "'edge set"   and x :: "'a" shows "CC E' x = connected E' `` {x}"
subgroups_of_size_def: fixes p :: "nat" shows "stabGroup.subgroups_of_size p = {H. subgroup H (G\<lparr>carrier := conj.stabilizer P\<rparr>) \<and> card H = p}"
distinct_29: fixes x2 :: "nat"   and x7 :: "rat" shows "Number x2 \<noteq> Pri_AgrK x7"
abs_ge_zero: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq zero (abs a)"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and c :: "'a"   and exp :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* c"   and "\<And>c. P2 c \<Gamma>"   and "\<And>c. P1 c exp" shows "P1 c (Let \<Gamma> exp)"
reachableI_1: fixes x :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and m :: "'g" assumes "x \<in> roots (s (mutator m))" shows "mut_m.reachable m x s"
JF1rel_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "JF1rel R = (BNF_Def.Grp (JF1in {(x, y). R x y}) (JF1map fst))\<inverse>\<inverse> OO BNF_Def.Grp (JF1in {(x, y). R x y}) (JF1map snd)"
cases: fixes ord :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a1 :: "'h option"   and a2 :: "'i option" assumes "ord_option ord a1 a2" obtains x :: "'i option" where "a1 = None" and "a2 = x__"   | x :: "'h" and y :: "'i" where "a1 = Some x" and "a2 = Some y" and "ord x y"
CondThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
abs_eq: fixes xb :: "'d pmf"   and xa :: "'d \<Rightarrow> 'c pmf"   and x :: "'d set" shows "bind_pmf_aux xb xa x = mapping.Mapping (\<lambda>xc. if xc \<in> (\<Union>y\<in>x. set_pmf (xa y)) then Some (measure_pmf.expectation xb (\<lambda>y. indicat_real x y * pmf (xa y) xc)) else None)"
lex_ext_unbounded_total: fixes ss :: "'b list"   and ts :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" assumes "\<forall>(s, t)\<in>set (zip ss ts). s = t \<or> fst (f s t) \<or> fst (f t s)"   and "\<And>t. snd (f t t)"   and "length ss = length ts" shows "ss = ts \<or> fst (lex_ext_unbounded f ss ts) \<or> fst (lex_ext_unbounded f ts ss)"
AL_Size_Zero: fixes L :: "('a \<times> 'b) list"   and a :: "'a" assumes "|L| = 0" shows "None = L\<down>a"
h'_outside: fixes x :: "'v"   and y :: "'v" assumes "(x, y) \<notin> E'" shows "h' x y = 0"
M_def: shows "M \<equiv> \<chi>i j. (if j = 1 then B else C) $ i - A $ i"
trivial_homomorphism_image: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd" assumes "Group.group G" shows "trivial_homomorphism G H f = (f \<in> hom G H \<and> f ` carrier G = {\<one>\<^bsub>H\<^esub>})"
sin_eq: fixes x :: "real"   and y :: "real" shows "(sin x = sin y) = (\<exists>n\<in>\<int>. x = y + 2 * n * pi \<or> x = - y + (2 * n + 1) * pi)"
lookup_pair_update_by_fun_pair: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k1 :: "'a"   and f :: "'c \<Rightarrow> 'c"   and k2 :: "'a" assumes "oalist_inv_raw x xs" shows "lookup_pair x (update_by_fun_pair x k1 f xs) k2 = (if k1 = k2 then f else id) (lookup_pair x xs k2)"
pointermap_p_pth_inv: fixes p :: "nat"   and m :: "'a pointermap"   and a :: "'a"   and x :: "nat"   and u :: "'a pointermap" assumes "pointermap_p_valid p m"   and "pointermap_getmk a m = (x, u)" shows "pm_pth u p = pm_pth m p"
set_less_eq_aux'_porder: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "partial_order_on infinite_complement_partition {(A, B). ord.set_less_eq_aux' less_eq A B}"
sep_disj_addI2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj (plus x z) y"
Compl_insert: fixes x :: "'g"   and A :: "'g set" shows "- insert x A = - A - {x}"
exhaust: fixes y :: "D" obtains x1 :: "int" where "y = A x1"   | x2 :: "char list" where "y = B x2"   | x3 :: "bool" where "y = C x3"
inject_3: fixes x4 :: "'f"   and y4 :: "'f" shows "(LSpawn x4 = LSpawn y4) = (x4 = y4)"
ex_mid_PL: fixes A :: "'a set"   and p :: "'a" shows "A \<turnstile> p .\<or>. .\<not> p"
exec_callFault: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and f :: "'f"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> \<Rightarrow> Fault f"
small_antisym: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> G * H"   and "h2 \<in> G * H"   and "h1 \<in> o[F](h2)"   and "h2 \<in> o[F](h1)" shows "False"
pred_fun_def: fixes A :: "'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> bool" shows "pred_fun A B = (\<lambda>f. \<forall>x. A x \<longrightarrow> B (f x))"
Zero_neq_One: shows "(0::'b) \<noteq> One"
mertens_third_convert: fixes n :: "nat" assumes "0 < n" shows "(\<Prod>k<n. 1 - 1 / real (nth_prime k)) = (\<Prod>x\<in>{p. prime p \<and> p \<le> nth_prime (n - 1)}. 1 - 1 / real x)"
set_2: fixes x21 :: "'f"   and x22 :: "'f list" shows "set (x21 # x22) = insert x21 (set x22)"
underS_UnderS_trans: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and C :: "'a set" assumes "trans r"   and "antisym r"   and "a \<in> underS r b"   and "b \<in> UnderS r C" shows "a \<in> UnderS r C"
abs_eq: fixes xa :: "'a \<Rightarrow> 'a"   and x :: "'a \<Rightarrow> 'a" assumes "eq_onp bij xa xa"   and "eq_onp bij x x" shows "Abs_permutation xa + Abs_permutation x = Abs_permutation (xa \<circ> x)"
pareto_losersI: fixes x :: "'b"   and R :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" assumes "x \<prec>[Pareto R] y" shows "x \<in> pareto_losers R"
int_sum_of_fourth_powers: fixes m :: "nat" shows "30 * int (\<Sum>i = 0..<m. i * i * i * i) = int m * (int m - 1) * (int (2 * m) - 1) * (int (3 * m * m) - int (3 * m) - 1)"
distinct_43: fixes x41 :: "'b poly"   and x42 :: "'b poly"   and x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly" shows "Sub x41 x42 \<noteq> CN x81 x82 x83"
assms_6: shows "a = (\<Otimes>x\<in>B. x [^] g x)"
rep_eq: fixes x :: "'b ttree" shows "paths x = Rep_ttree x"
v: shows "v \<in> {v. [v ^ CARD('a) = v] (mod u)}"
plusl_num_infty: fixes x :: "nat" assumes "0 < x" shows "[x]\<^sub>1 +\<^sub>1 \<infinity>\<^sub>1 = \<infinity>\<^sub>1"
not_strict_par1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p" assumes "A B Par C D"   and "Col A B X"   and "Col C D X" shows "Col A B C"
splus_prod_def: fixes t :: "'a \<times> 'b"   and u :: "'a \<times> 'b" shows "splus t u = pprod.splus (fst t) u"
elims_3: fixes x :: "'b status" assumes "\<not> st_inv x" obtains ok :: "nat" and f :: "'b list" and f' :: "'b list" and r :: "'b list" and r' :: "'b list" where "x = Rev ok__ f f'__ r r'__" and "\<not> (length f + 1 = length r \<and> length f'__ = length r'__ \<and> ok__ \<le> length f'__)"   | ok :: "nat" and f' :: "'b list" and r' :: "'b list" where "x = App ok f' r'" and "\<not> (ok \<le> length f' \<and> length f' < length r')"
UN_least: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and C :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C" shows "\<Union> (B ` A) \<subseteq> C"
strict_coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<sqsubset> c" shows "min a b \<sqsubset> c"
assms: fixes x :: "'a"   and Q :: "'a set" assumes "x \<in> Q"   and "Q \<subseteq> A" shows "\<exists>z\<in>Q. \<forall>y\<in>A. r y z \<longrightarrow> y \<notin> Q"
aprimedivisor_le_nat: fixes n :: "nat" assumes "Suc 0 < n" shows "aprimedivisor n \<le> n"
snd_vimage_eq_Times: fixes S :: "'c set" shows "snd -` S = UNIV \<times> S"
get_element_by_id_is_strongly_scdom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and idd :: "char list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_element_by_id ptr' idd \<rightarrow>\<^sub>r Some result" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set sc) = (ptr' \<in> set sc)"
unit_simps_1: shows "S.seq (S.inv (S.cmp\<^sub>U\<^sub>P (g, f))) (S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f))"
inner_lv_rel_def: fixes a :: "'a list"   and b :: "'a list" shows "inner_lv_rel a b = sum_list (map2 (*) a b)"
nBnd: shows "nBnd = fact (n + 1) * max (1::'a) Bnd ^ n"
norm2_le_norm1_int: fixes f :: "int poly" shows "\<parallel>f\<parallel>\<^sup>2 \<le> (norm1 f)\<^sup>2"
assms_2: fixes a :: "'a" assumes "a \<in> set_pmf M"   and "f a \<noteq> 0" shows "a \<in> A"
mat_of_cblinfun_Proj_code_def: fixes S :: "'a ccsubspace" shows "mat_of_cblinfun_Proj_code S = mat_of_cblinfun (Proj S)"
left_hom_axioms: shows "left_hom (\<cdot>) (\<star>) (trg f)"
rbt_comps_2: shows "rbt_comp_lookup ccomp = local.rbt_lookup"
simps_2: fixes v :: "nat"   and s :: "'b sparc_state" shows "SEQ (Suc v) s = (case SEQ (Suc v - 1) s of None \<Rightarrow> None | Some x \<Rightarrow> NEXT x)"
divide_rat_def: fixes q :: "rat"   and r :: "rat" shows "q / r = q * inverse r"
iter_bind_iter: fixes m :: "nat"   and s :: "'a"   and n :: "nat" shows "iter m s \<bind> iter n = iter (m + n) s"
safe_config: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and n :: "nat"   and safe :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool"   and c\<^sub>0 :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and k :: "nat"   and c :: "nat \<Rightarrow> ('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and l :: "nat" assumes "program_safe_reach_upto program_step n safe c\<^sub>0"   and "k \<le> n"   and "program_trace program_step c 0 k"   and "c 0 = c\<^sub>0"   and "l \<le> k" shows "safe (c l)"