hcju/isabelleps huggingface dataset:
gcd_le2_nat: fixes b :: "nat"   and a :: "nat" assumes "b \<noteq> 0" shows "gcd a b \<le> b"
diamond_parcontract: shows "diamond (\<Rrightarrow>\<^sup>1)"
UNION_in_sets: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and A :: "nat \<Rightarrow> 'b set"   and n :: "nat" assumes "ring_of_sets \<Omega> M"   and "range A \<subseteq> M" shows "\<Union> (A ` {0..<n}) \<in> M"
listassoc_sym: fixes as :: "'a list"   and bs :: "'a list" assumes "as [\<sim>] bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "bs [\<sim>] as"
uint_word_of_int_bitlen_eq: fixes x :: "int" assumes "bitlen x \<le> int LENGTH('a)"   and "0 \<le> x" shows "uint (word_of_int x) = x"
Uniq_I: fixes P :: "'e \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> y = x" shows "Uniq P"
not_integrable_integral: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a set" assumes "\<not> f integrable_on i" shows "integral i f = (0::'b)"
neq_if_length_neq: fixes xs :: "'k list"   and ys :: "'k list" assumes "length xs \<noteq> length ys" shows "xs = ys \<equiv> False"
cfun_comp_def: shows "cfun_comp = (\<Lambda> f \<rho>. Rep_cfun f \<circ> \<rho>)"
simps_4: fixes r :: "mregex"   and s :: "mregex" shows "RPD (MPlus r s) = RPD r \<union> RPD s"
interfree_swap_def: shows "interfree_swap \<equiv> \<lambda>(x, xs). \<forall>y\<in>set xs. interfree_aux (OG_Hoare.com x, OG_Hoare.post x, OG_Hoare.com y) \<and> interfree_aux (OG_Hoare.com y, OG_Hoare.post y, OG_Hoare.com x)"
dmb_le_dbm_entry_bound_inf: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<preceq> b"   and "a = \<infinity>" shows "b = \<infinity>"
Z1: shows "Esigma (2 ^ k * p * q) = 2 ^ k * p * q + 2 ^ k * r"
chain_subdiv_chain_imp_finite_subdiv: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "finite one_chain1"   and "chain_subdiv_chain one_chain1 subdiv" shows "finite subdiv"
scale_zero_left: fixes x :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.mult_scalar 0 x = 0"
add_inc: fixes x :: "num"   and y :: "num" shows "x + Num.inc y = Num.inc (x + y)"
Least_eq_0: fixes P :: "nat \<Rightarrow> bool" assumes "P 0" shows "Least P = 0"
nat_le_iff_add: fixes m :: "nat"   and n :: "nat" shows "((m::nat) \<le> (n::nat)) = (\<exists>k::nat. n = m + k)"
iso_verts_eq_iff: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'a"   and v :: "'a" assumes "digraph_isomorphism hom"   and "u \<in> verts G"   and "v \<in> verts G" shows "(iso_verts hom u = iso_verts hom v) = (u = v)"
openin_continuous_map_preimage: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'d set" assumes "continuous_map X Y f"   and "openin Y U" shows "openin X {x \<in> topspace X. f x \<in> U}"
not_na_ge_inf: fixes n :: "nat" shows "\<not> \<infinity> \<le> an n"
tr_valid_preserve: fixes c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and c' :: "'n list multiset" assumes "(c, w, c') \<in> trcl (tr fg)"   and "valid fg c" shows "valid fg c'"
fimage_is_vempty: fixes r :: "V"   and A :: "V" shows "(r `\<^sub>\<bullet> A = []\<^sub>\<circ>) = (\<D>\<^sub>\<bullet> r \<inter>\<^sub>\<circ> A = []\<^sub>\<circ>)"
freshAbs_psubstAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A"   and "goodEnv rho" shows "freshAbs zs z A $[rho] = (\<forall>ys y. freshAbs ys y A \<or> freshImEnvAt zs z rho ys y)"
C2: fixes y :: "name"   and \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and zvec :: "name list"   and S :: "('a, 'b, 'c) psi" assumes "y \<sharp> \<Psi>'"   and "y \<sharp> R"   and "zvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*zvec\<rparr>R \<parallel> \<lparr>\<nu>y\<rparr>S, \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>*zvec\<rparr>R \<parallel> S)) \<in> Rel"
resource_of_oracle_parametric: fixes S :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun A (rel_spmf (rel_prod B S)))) (rel_fun S (rel_resource A B)) resource_of_oracle resource_of_oracle"
simps: fixes fv1 :: "'b \<Rightarrow> atom set"   and fv2 :: "'c \<Rightarrow> atom set"   and x :: "'b"   and y :: "'c" shows "prod_fv fv1 fv2 (x, y) = fv1 x \<union> fv2 y"
Cauchy_uniform: fixes "open" :: "'b set \<Rightarrow> bool"   and uniformity :: "('b \<times> 'b) filter"   and X :: "nat \<Rightarrow> 'b" assumes "class.uniform_space open uniformity" shows "uniform_space.Cauchy uniformity X = uniform_space.cauchy_filter uniformity (filtermap X sequentially)"
filterlim_tendsto_pos_mult_at_top_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c" shows "(LIM x F. f x * g x :> at_top) = filterlim g at_top F"
hom_into_subgroup_eq_gen: fixes G :: "('a, 'b) monoid_scheme"   and h :: "'c \<Rightarrow> 'a"   and K :: "('c, 'd) monoid_scheme"   and H :: "'a set" assumes "Group.group G" shows "(h \<in> hom K (subgroup_generated G H)) = (h \<in> hom K G \<and> h ` carrier K \<subseteq> carrier (subgroup_generated G H))"
measurable_fun_upd_state_measure: fixes v :: "nat"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "v \<notin> V" shows "(\<lambda>(x, y). y(v := x)) \<in> stock_measure (\<Gamma> v) \<Otimes>\<^sub>M state_measure V \<Gamma> \<rightarrow>\<^sub>M state_measure (insert v V) \<Gamma>"
widen_ivl_def: fixes ivl1 :: "ivl"   and ivl2 :: "ivl" shows "ivl1 \<nabla> ivl2 = (case (ivl1, ivl2) of (I l1 h1, I l2 h2) \<Rightarrow> I (if le_option False l2 l1 \<and> l2 \<noteq> l1 then None else l1) (if le_option True h1 h2 \<and> h1 \<noteq> h2 then None else h1))"
var_stmt_inj_term: fixes t :: "Term.var"   and w :: "stmt" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
star_rtc2: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "(t_one \<union> t_prod x x \<subseteq> x) = (x = t_star x)"
stmt_expr_inj_term: fixes t :: "stmt"   and w :: "expr" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
a1: shows "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus"
borel_measurable_divide: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and g :: "'d \<Rightarrow> 'e" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
affine_def: fixes s :: "'b set" shows "affine s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
vlist_of_vfsequence_vfsequence_of_vlist: fixes xs :: "V list" shows "vlist_of_vfsequence (vfsequence_of_vlist xs) = xs"
reachable_def: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c" shows "Graph_Start_Defs.reachable E s\<^sub>0 = E\<^sup>*\<^sup>* s\<^sub>0"
assm_2: fixes x :: "'b"   and K :: "'b set" assumes "(x, K) \<in> p" shows "x \<in> K"
assms_4: shows "2 ^ card (prob_dom PROB) - 1 < length as"
weaken_left_insert: fixes G :: "'b pl set"   and p :: "'b pl"   and a :: "'b pl" assumes "G |- p" shows "insert a G |- p"
assms_1: shows "(\<exists>\<^sub>cx. \<gamma> x) t n"
head_Var_reduction: fixes n :: "name"   and rs :: "lam list"   and v :: "lam" assumes "Var n \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> v" shows "\<exists>ss. rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss \<and> v = Var n \<degree>\<degree> ss"
uses'_code_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val"   and n :: "'node" shows "uses'_code g next n \<equiv> substNext_code g next ` uses g n"
load_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t1 :: "'b sparc_state"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> (fst instr = load_store_type LDSB \<or> fst instr = load_store_type LDUB \<or> fst instr = load_store_type LDUH \<or> fst instr = load_store_type LD \<or> fst instr = load_store_type LDD) \<and> t1 = snd (fst (load_sub1 instr rd 0 s1)) \<and> t2 = snd (fst (load_sub1 instr rd 0 s2))" shows "low_equal t1 t2"
r_code_constn: fixes n :: "nat"   and c :: "nat" shows "eval (r_code_constn n) [c] \<down>= code_constn n c"
suntil_Stream: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and \<psi> :: "'c stream \<Rightarrow> bool"   and x :: "'c"   and s :: "'c stream" shows "(\<phi> suntil \<psi>) (x ## s) = (\<psi> (x ## s) \<or> \<phi> (x ## s) \<and> (\<phi> suntil \<psi>) s)"
div_mult2_numeral_eq: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b"   and k :: "num"   and l :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (divide a (numeral.numeral one plus k)) (numeral.numeral one plus l) = divide a (numeral.numeral one plus (k * l))"
simps: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float list"   and f :: "floatarith" shows "tm_floatarith prec ord I a f = map_option (\<lambda>cs. let (pf, pi) = tmf_polys cs; _ = compute_bound_tm prec (map2 (-) I (map interval_of a)) in Let (round_interval prec (Ipoly (map2 (-) I (map interval_of a)) pi)) (TaylorModel pf)) (tmf_ivl_cs prec ord I a f)"
ltl_prop_equiv_trans: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<chi> :: "'a ltln" assumes "\<phi> \<sim>\<^sub>P \<psi>"   and "\<psi> \<sim>\<^sub>P \<chi>" shows "\<phi> \<sim>\<^sub>P \<chi>"
dyad_rec_level_termination: fixes k :: "nat"   and K :: "nat"   and b :: "nat \<Rightarrow> 'b"   and l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and m :: "nat" assumes "k < K" shows "dyad_rec_dom (b, l, r, real m / 2 ^ k)"
SeqHRP_HPair: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and x :: "tm"   and y :: "tm"   and x' :: "tm"   and y' :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, x, y, x', y')"   and "atom k \<sharp> (s1, s2, k1, k2, x, y, x', y')" shows "{SeqHRP x x' s1 k1, SeqHRP y y' s2 k2} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqHRP (HPair x y) (Q_HPair x' y') (Var s) (Var k)))"
opnet_cast1: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and R :: "nat set"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and H :: "nat set"   and K :: "nat set"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), R:*cast(m), \<sigma>', s') \<in> S"   and "((\<sigma>, t), H\<not>K:arrive(m), \<sigma>', t') \<in> T"   and "H \<subseteq> R"   and "K \<inter> R = {}" shows "((\<sigma>, SubnetS s t), R:*cast(m), \<sigma>', SubnetS s' t') \<in> opnet_sos S T"
stable_on_def: fixes ds :: "'d set"   and X :: "'x set" shows "stable_on ds X = (individually_rational_on ds X \<and> stable_no_blocking_on ds X)"
einterval_eq_Icc: fixes a :: "real"   and b :: "real" shows "einterval (ereal a) (ereal b) = {a<..<b}"
elims: fixes x :: "'b bro"   and y :: "'b bro" assumes "tree x = y" obtains t :: "'b bro" where "x = N1 t" and "y = t"   | "x = N0" and "y = N0"   | v :: "'b bro" and va :: "'b" and vb :: "'b bro" where "x = N2 v__ va__ vb__" and "y = N2 v__ va__ vb__"   | v :: "'b" where "x = L2 vc__" and "y = L2 vc__"   | v :: "'b bro" and va :: "'b" and vb :: "'b bro" and vc :: "'b" and vd :: "'b bro" where "x = N3 v va vb vc vd" and "y = N3 v va vb vc vd"
roomk_issued: fixes s :: "state"   and k :: "room" assumes "s \<in> reach" shows "roomk s k \<in> issued s"
open_halfspace_Im_gt: fixes b :: "real" shows "open {z. b < Im z}"
ivl_disj_int_two_3: fixes l :: "'a"   and m :: "'a"   and u :: "'a" shows "{l..<m} \<inter> {m..<u} = {}"
remdupsAppendSet: fixes a :: "'b list"   and b :: "'b list" shows "set (remdups (a @ b)) = set (remdups a @ remdups (list_diff b a))"
function_one_is_constant: shows "\<cc>\<^bsub>Zp\<^esub> \<one> = \<one>\<^bsub>function_ring (carrier Zp) Zp\<^esub>"
singular_simplex_def: fixes p :: "nat"   and X :: "'c topology"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" shows "singular_simplex p X f \<equiv> continuous_map (subtopology (powertop_real UNIV) (standard_simplex p)) X f \<and> f \<in> extensional (standard_simplex p)"
1: shows "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) contour_integrable_on \<gamma>"
Span_contains_spanset: fixes ms :: "'b list" assumes "set ms \<subseteq> V" shows "set ms \<subseteq> RSpan ms"
assms_4: fixes i :: "'a" assumes "i \<in> Basis" shows "convex (X i)"
select_convs_2: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.funcs \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = funcs"
Int_absorb1: fixes B :: "'b set"   and A :: "'b set" assumes "B \<subseteq> A" shows "A \<inter> B = B"
sigma_def: shows "sigma \<equiv> \<lambda>A. {x. sigmap (\<lambda>x. x \<in> A) x}"
bij_betw_arr_sets: shows "bij_betw \<phi> (Collect J'J.A.arr) (Collect J.arr)"
Suc_2: shows "\<forall>k'\<le>n. \<one> \<le> fw m n k n n k' k'"
Image_Un: fixes R :: "('b \<times> 'a) set"   and A :: "'b set"   and B :: "'b set" shows "R `` (A \<union> B) = R `` A \<union> R `` B"
cases: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and a :: "'e" assumes "terminates_onp g a" obtains s :: "'e" where "a = sb__" and "g sb__ = Done"   | s :: "'e" and s' :: "'e" where "a = sc__" and "g sc__ = Skip s'a__" and "terminates_onp g s'a__"   | s :: "'e" and aa :: "'f" and s' :: "'e" where "a = s" and "g s = Yield aa s'" and "terminates_onp g s'"
assms_2: shows "X \<subseteq>\<^sub>L Y"
trace_callee_eqD: fixes callee1 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and callee2 :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and A :: "'b set"   and p :: "'a spmf"   and q :: "'d spmf"   and xs :: "('b \<times> 'c) list"   and x :: "'b" assumes "trace_callee_eq callee1 callee2 A p q"   and "set xs \<subseteq> A \<times> UNIV"   and "x \<in> A" shows "trace_callee callee1 p xs x = trace_callee callee2 q xs x"
table_empty: fixes n :: "nat"   and X :: "'a option list set" shows "table n {} X = (X = empty_table \<or> X = unit_table n)"
poly_eval_sum: fixes a :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and I :: "'d set" shows "poly_eval a (sum f I) = (\<Sum>i\<in>I. poly_eval a (f i))"
homotopic_paths_linv: fixes p :: "real \<Rightarrow> 'b"   and s :: "'b set" assumes "path p"   and "path_image p \<subseteq> s" shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
bind_rzero: fixes x :: "'d option" shows "x \<bind> Map.empty = None"
simps_9: fixes v1 :: "'b"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (V v1) (Minus v va) = True"
fin_rest_elims_1: fixes a :: "coname"   and M :: "trm"   and x :: "name"   and N :: "trm"   and y :: "name" assumes "fin (Cut <a>.M x.N) y" shows "False"
matrix_match_def: fixes A1 :: "int poly fract list list"   and A2 :: "int poly fract list list"   and B1 :: "int poly fract list list"   and B2 :: "int poly fract list list" shows "rat_poly.matrix_match A1 A2 B1 B2 \<equiv> mat (rat_poly.row_length A1) (length A1) A1 \<and> mat (rat_poly.row_length A2) (length A2) A2 \<and> mat (rat_poly.row_length B1) (length B1) B1 \<and> mat (rat_poly.row_length B2) (length B2) B2 \<and> length A1 = rat_poly.row_length A2 \<and> length B1 = rat_poly.row_length B2 \<and> A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []"
mult_zero_2: fixes f :: "nat \<Rightarrow> int" assumes "f \<in> carrier Zp" shows "\<zero> \<otimes> f = \<zero>"
memoized_correct: fixes os :: "'t iarray"   and a :: "'s"   and b :: "'s"   and c :: "nat" shows "forward_ix_rec os a b c = fst (run_state (forward_ix\<^sub>m' os a b c) Mapping.empty)"
add_less_same_cancel2: fixes a :: "'a"   and b :: "'a" shows "(a + b < b) = (a < (0::'a))"
g: fixes K :: "'c set" assumes "compactin Z K" shows "compactin Y {x \<in> topspace Y. g x \<in> K}"
exec_gpv_inline: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and gpv :: "'c \<Rightarrow> 'g \<Rightarrow> ('h \<times> 'c, 'e, 'f) gpv"   and c' :: "('b, 'g, 'h) gpv"   and s' :: "'c"   and s :: "'d" shows "exec_gpv callee (inline gpv c' s') s = map_spmf (\<lambda>(x, s', s). ((x, s'), s)) (exec_gpv (\<lambda>(s', s) y. map_spmf (\<lambda>((x, s'), s). (x, s', s)) (exec_gpv callee (gpv s' y) s)) c' (s', s))"
step: fixes cfg1 :: "('p, 'v, 's) configuration"   and cfg2 :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message"   and cfg3 :: "('p, 'v, 's) configuration" assumes "reachable cfg1 cfg2"   and "cfg2 \<turnstile> msg \<mapsto> cfg3" shows "reachable cfg1 cfg3"
le_PlusL: fixes r :: "'a language"   and s :: "'a language" shows "r \<le> Plus r s"
AE_restr_to_subalg2: fixes M :: "'b measure"   and F :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "subalgebra M F"   and "almost_everywhere M P"   and "Measurable.pred F P" shows "almost_everywhere (restr_to_subalg M F) P"
util_imp_refl: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "refl_on carrier relation"
fresh_2: fixes a :: "name"   and x1 :: "name" shows "a \<sharp> BoundOutputS x1 = a \<sharp> x1"
add_edge_correct_2: fixes \<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and \<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and invar :: "'a \<Rightarrow> bool"   and adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and empty :: "'a"   and add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and u :: "'b"   and v :: "'b"   and d :: "nat" assumes "adt_wgraph \<alpha>w \<alpha>g invar adj empty add_edge"   and "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "\<alpha>g (add_edge (u, v) d g) = ins_edge (u, v) (\<alpha>g g)"
indicator_eq_1_iff: fixes A :: "'d set"   and x :: "'d" shows "(indicator A x = (1::'c)) = (x \<in> A)"
axioms_1: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subdomain H R" shows "subcring H R"
converse_trancl_induct: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (y, b) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>+; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
function_times_one_r: fixes a :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier F" shows "a \<otimes>\<^bsub>F\<^esub> \<one>\<^bsub>F\<^esub> = a"
unique_PPc_def: fixes c :: "('a, 'b) MWLsCom" shows "unique_PPc c = distinct (PPc c)"
enc_alt_def: fixes w :: "'a list"   and x :: "nat + nat set"   and I :: "(nat + nat set) list" shows "enc (w, x # I) = map_index (\<lambda>n (a, bs). (a, (case x of Inl p \<Rightarrow> n = p | Inr P \<Rightarrow> n \<in> P) # bs)) (enc (w, I))"
path_Entry_target_2: fixes n :: "'node"   and as :: "'edge list" assumes "n -as\<rightarrow>* (_Entry_)" shows "as = []"
insert_DiffM2: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "M - (x, {#}) + (x, {#}) = M"
exec_step_imp_exec_step_ind: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and xp' :: "nat option"   and h' :: "heap"   and frs' :: "frame list"   and sh' :: "sheap" assumes "exec_step P h stk loc C M pc ics frs sh = (xp', h', frs', sh')" shows "exec_step_ind (exec_step_input P C M pc ics) P h stk loc C M pc ics frs sh (xp', h', frs', sh')"
ptr_kinds_M_ok: fixes h :: "(_) heap" shows "h \<turnstile> ok document_ptr_kinds_M"
assms_2: shows "is_parallel_solution_for_problem \<Pi> \<pi>"
absorb_iff1: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a \<squnion> b = a)"
C: shows "set (y # ys) \<subseteq> range p \<union> range q"
Replace_def: fixes A :: "hf"   and R :: "hf \<Rightarrow> hf \<Rightarrow> bool" shows "Replace A R = PrimReplace A (\<lambda>x y. (\<exists>!z. R x z) \<and> R x y)"
inject_3: fixes x5 :: "loc"   and y5 :: "loc" shows "(Addr x5 = Addr y5) = (x5 = y5)"
Increasing_def: fixes f :: "'c \<Rightarrow> 'd" shows "Increasing f \<equiv> \<Inter>z. Stable {s. z \<le> f s}"
a_closed: fixes x :: "'a"   and y :: "'a" shows "d (ad x \<cdot> ad y) = ad x \<cdot> ad y"
interior_atLeastAtMost_real: fixes a :: "real"   and b :: "real" shows "interior {a..b} = {a<..<b}"
bigI: fixes c :: "real"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "0 < c"   and "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)" shows "f \<in> O[F](g)"
simps_6: fixes x3 :: "tnam" shows "TName x3 \<noteq> Object'"
multi_nonempty_split: fixes M :: "'a multiset" assumes "M \<noteq> {#}" shows "\<exists>A a. M = add_mset a A"
eval_is_hom: fixes K :: "'a list set"   and a :: "'a list" assumes "subring K (poly_ring R)"   and "a \<in> carrier (poly_ring R)" shows "(\<lambda>p. UP.eval p a) \<in> ring_hom (K [X]\<^bsub>poly_ring R\<^esub>) (poly_ring R)"
open_contains_cball: fixes S :: "'c set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
cart_product_add_1_filter_eq: fixes a :: "'a"   and M :: "'a multiset"   and N :: "'a set multiset" shows "{#m \<in># add_mset a M \<times># N. fst m \<in> snd m#} = {#m \<in># M \<times># N. fst m \<in> snd m#} + {#a#} \<times># filter_mset ((\<in>) a) N"
cut_greater_def: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t \<equiv> {x \<in> I. t < x}"
status_def: shows "status = {(HA, C, E, D) |HA C E D. Status HA C E D}"
independent_and_consistent_iarrays_def: fixes A :: "'d iarray iarray"   and b :: "'d iarray" shows "independent_and_consistent_iarrays A b = (let GJ = Gauss_Jordan_iarrays_PA A; rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (snd GJ))); P_mult_b = fst GJ *iv b; consistent_A = (if \<not> is_zero_iarray P_mult_b then greatest_not_zero P_mult_b + 1 else 0) \<le> rank_A; dim_solution_set = ncols_iarray A - rank_A in consistent_A \<and> dim_solution_set = 0)"
sel_6: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.LI (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x6"
\<delta>_mk_fun: fixes Ts :: "Unary_PCF.type list"   and T :: "Unary_PCF.type" shows "\<delta> (Ts \<rightarrow>\<rightarrow> T) = HMSet (hmsetmset (\<delta> T) + mset (map \<delta> Ts))"