hcju/isabelleps huggingface dataset:
spannable_def: fixes x :: "'a" shows "pd_kleene_allegory.spannable (\<lambda>x. top) top (\<sqinter>) (\<squnion>) (\<le>) id uminus x \<equiv> \<exists>y\<le>x. x \<le> top \<and> y \<sqinter> top \<le> - top \<and> y \<sqinter> id y \<le> top"
crename_NotR': fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and N :: "trm"   and c :: "coname" assumes "R[a\<turnstile>c>b] = NotR x.N c"   and "x \<sharp> R"   and "c \<sharp> a" shows "(\<exists>N'. R = NotR x.N' c \<and> N'[a\<turnstile>c>b] = N) \<or> (\<exists>N'. R = NotR x.N' a \<and> b = c \<and> N'[a\<turnstile>c>b] = N)"
all_equivalent: fixes x :: "'c"   and y :: "'c" assumes "x \<in> E"   and "y \<in> E" shows "x .=\<^bsub>\<lparr>carrier = E, eq = \<lambda>x y. y \<in> orbit G \<phi> x\<rparr>\<^esub> y"
method: shows "P\<^sub>1 \<turnstile> C sees M: Ts\<rightarrow>T = body in C"
produce_nil: fixes ts :: "checker_type" shows "produce ts (Type []) = ts"
norm_g: shows "\<forall>\<^sub>F x in F. norm (g x) \<le> B"
finite_dual: shows "finite \<circ> (`) \<partial> = finite"
contlub_cfun_arg: fixes Y :: "nat \<Rightarrow> 'b"   and f :: "'b \<rightarrow> 'c" assumes "chain Y" shows "f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
assms: shows "is_density_expr (vs, vs', \<Gamma>, \<delta>) t e"
ordered_fmap_remove: fixes S :: "('a \<times> 'b) fset"   and x :: "'a \<times> 'b" assumes "is_fmap S"   and "x |\<in>| S" shows "ordered_fmap (S |-| {|x|}) = remove1 x (ordered_fmap S)"
\<ll>_ide_simp: fixes f :: "'b" assumes "ide f" shows "\<ll> f = \<l>[f]"
Substable_def: shows "Substable \<equiv> \<lambda>cond \<phi>. \<forall>\<psi> \<chi> v. cond \<psi> \<chi> \<longrightarrow> [\<phi> \<psi> \<^bold>\<equiv> \<phi> \<chi> in v]"
norm_scaleR: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and a :: "real"   and x :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (scaleR a x) = \<bar>a\<bar> * norm x"
intros_2: fixes t :: "('b, 'c, 'd) Tree" shows "(t, tNot t) \<in> Tree_wf"
big_small_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g"   and "g \<in> l F h" shows "f \<in> l F h"
nextMax_Un_eq: fixes x :: "nat set"   and y :: "nat set" assumes "finite x"   and "finite y" shows "nextMax (x \<union> y) = max (nextMax x) (nextMax y)"
FE_IMP_RED_PRE: fixes blues0 :: "'v set"   and reds0 :: "'v set"   and onstack0 :: "'v set"   and u0 :: "'v"   and blues :: "'v set"   and reds :: "'v set"   and onstack :: "'v set" assumes "pre (blues0, reds0, onstack0, u0)"   and "fe_inv (insert u0 blues0) u0 (insert u0 onstack0) {} (blues, reds, onstack, NO_CYC)"   and "u0 \<in> F" shows "u0 \<notin> reds"
update_by_fun_gr_pair_eq_update_by_fun_pair: fixes xs :: "('a \<times> 'b) list"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "tc.oalist_inv_raw xs" shows "update_by_fun_gr_pair_tc k f xs = update_by_fun_pair_tc k f xs"
PDQ_diag: shows "(P'', D, Q'') = diagonalize_JNF A bezout"
mult_stalk_def: shows "stk1.mult_stalk \<equiv> stk1.mult_rel"
Ax_subset_false_aux: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "A \<inter> B = {}"   and "A \<noteq> {}" shows "False"
continuous_map_cases_le: fixes X :: "'a topology"   and p :: "'a \<Rightarrow> real"   and q :: "'a \<Rightarrow> real"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_map X euclideanreal p"   and "continuous_map X euclideanreal q"   and "continuous_map (subtopology X {x \<in> topspace X. p x \<le> q x}) Y f"   and "continuous_map (subtopology X {x \<in> topspace X. q x \<le> p x}) Y g"   and "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x" shows "continuous_map X Y (\<lambda>x. if p x \<le> q x then f x else g x)"
fls_base_factor_const: fixes c :: "'a" shows "fls_base_factor (fls_const c) = fls_const c"
butlast_take_length_helper: fixes i2 :: "char list"   and i1 :: "char list"   and x :: "char list" assumes "length i2 \<le> length i1"   and "take (length i2 - Suc 0) i1 = butlast i2"   and "butlast i1 = take (length i1 - Suc 0) x" shows "butlast i2 = take (length i2 - Suc 0) x"
vertical_composite_axioms: shows "vertical_composite A B G F G \<sigma>' \<sigma>"
CastRedsFail: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s',b'\<rangle>"   and "hp s' a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>Cast C e,s,b\<rangle> \<rightarrow>* \<langle>THROW ClassCast,s',b'\<rangle>"
of_int_sum: fixes one :: "'b"   and times :: "'b \<Rightarrow> '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 f :: "'c \<Rightarrow> int"   and A :: "'c set" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (sum f A) = comm_monoid_add.sum plus zero (\<lambda>x. ring_1.of_int one plus zero minus (f x)) A"
restr_stack_map_Dummy: fixes V :: "Vars.var set"   and l :: "Vars.var list" shows "restr_stack V (map Dummy l) = map Dummy l"
changes_nonneg: fixes xs :: "'a list" shows "0 \<le> changes xs"
real_power_1_right_aux: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and d :: "real" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < d" shows "real_power d one = scaleR d one"
is_process_REP_Skip: shows "is_process ({(s, X). s = [] \<and> tick \<notin> X} \<union> {(s, X). s = [tick]}, {})"
dim_eq_full: fixes S :: "('a, 'b) vec set" shows "(vec.dim S = finite_dimensional_vector_space.dimension cart_basis) = (vec.span S = UNIV)"
obind_return: fixes m :: "'b \<Rightarrow> 'c option" shows "m |>> oreturn = m"
override_on_insert': fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and X :: "'c set" shows "f ++\<^bsub>insert x X\<^esub> g = f(x := g x) ++\<^bsub>X\<^esub> g"
simps: fixes N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and n :: "nat" shows "is_final_dstate (N, P, Q, n) = (N = [] \<and> P = [])"
run_V0: fixes G :: "('c, 'd) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'c" assumes "graph_defs.is_run G r" shows "r 0 \<in> g_V0 G"
bisimScopeExtSym: fixes x :: "name"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>"   and "x \<sharp> Q" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> \<lparr>\<nu>x\<rparr>P \<parallel> Q"
decreasePower_insertion: fixes v :: "nat"   and p :: "real mpoly"   and p' :: "real mpoly"   and i :: "nat"   and f :: "nat \<Rightarrow> real" assumes "decreasePower v p = (p', i)" shows "insertion f p = insertion f p' * f v ^ i"
two_powr_nat_float: fixes i :: "nat" shows "2 powr real i \<in> float"
removeAll_mset_filter_mset: fixes M :: "'b multiset"   and C :: "'b" shows "removeAll_mset C M = filter_mset ((\<noteq>) C) M"
sets_dist_measure: fixes x :: "val"   and dst :: "pdf_dist" assumes "val_type x = dist_param_type dst" shows "sets (dist_measure dst x) = sets (stock_measure (dist_result_type dst))"
arr_Set_ArrVal_vdomain: shows "\<D>\<^sub>\<circ> (F\<lparr>ArrVal\<rparr>) = A"
meet_top: fixes P :: "'a hrel" assumes "P is \<H>" shows "P \<sqinter> \<^bold>\<top> = P"
ilen: shows "i < length (conc.restrict_modes mdss (- range var\<^sub>C_of))"
Domainp_iff: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e" shows "Domainp T x = (\<exists>y. T x y)"
intruder_deduct_hom_def: fixes M :: "('fun, 'var) Term.term set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" shows "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t \<equiv> \<langle>M;\<lambda>t. homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec \<and> t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)\<rangle> \<turnstile>\<^sub>r t"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "a \<preceq> b" shows "a \<prec> b"
trading_strategy_def: fixes p :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" shows "trading_strategy p = (portfolio p \<and> (\<forall>asset\<in>support_set p. borel_predict_stoch_proc F (p asset)))"
inf_assn_def: fixes P :: "assn"   and Q :: "assn" shows "inf P Q \<equiv> Abs_assn (\<lambda>h. h \<Turnstile> P \<and> h \<Turnstile> Q)"
OAlist_tc_prod_ord_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "('a, 'b) oalist_tc"   and ys :: "('a, 'c) oalist_tc" shows "OAlist_tc_prod_ord f xs ys = prod_ord_pair_tc f (list_of_oalist_tc xs) (list_of_oalist_tc ys)"
elims_2: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "('d, 'e, 'f) FSM" assumes "x \<preceq> xa" obtains M1 :: "('d, 'e, 'f) FSM" and M2 :: "('d, 'e, 'f) FSM" where "x = M1" and "xa = M2" and "L M1 \<subseteq> L M2"
ptr_kinds_M_ok: fixes h :: "'a"   and ptr_kinds :: "'a \<Rightarrow> 'b fset" shows "h \<turnstile> ok l_ptr_kinds_M.a_ptr_kinds_M ptr_kinds"
lin_independent_ConsD1: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b"   and ms :: "'b list" assumes "R_scalar_mult.R_lin_independent UNIV smult (m # ms)" shows "R_scalar_mult.R_lin_independent UNIV smult ms"
renumber: fixes a :: "'d"   and f :: "nat \<Rightarrow> 'd"   and r :: "('d \<times> 'd) set" assumes "\<forall>i. (a, f i) \<in> r\<^sup>* \<and> (f i, f (Suc i)) \<in> r" shows "\<exists>f. f 0 = a \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)"
quotient_of_number_1: shows "quotient_of 0 = (0)\<^sub>N"
Inf_insert: fixes a :: "'a"   and A :: "'a set" shows "\<Sqinter> (insert a A) = a \<sqinter> \<Sqinter> A"
r1: shows "\<rho>1 \<approx> (y, w1) # \<rho>"
of_nat_zero_less_power_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less zero (power.power one times (semiring_1.of_nat one plus zero x) n) = (0 < x \<or> n = 0)"
right_mult_one_mat': fixes A :: "'a Matrix.mat"   and n :: "nat" assumes "dim_col A = n" shows "A * 1\<^sub>m n = A"
tdev'_uminus_pdevs: fixes p :: "nat"   and x :: "'a pdevs" shows "tdev' p (uminus_pdevs x) = tdev' p x"
flowD_finite_IN: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "flow \<Delta> f"   and "x \<noteq> sink \<Delta>" shows "d_IN f x \<noteq> \<top>"
mapOf_lookup1: fixes i :: "int"   and t :: "(int \<times> 'b) Tree" assumes "tlookup fst i t = None" shows "mapOf t i = None"
acyc: shows "acyclicP (\<lambda>y x. y \<in># Y \<and> x \<in># Y \<and> gt y x)"
smap_eq_scons: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c stream"   and y :: "'b"   and ys :: "'b stream" shows "(smap f xs = y ## ys) = (f (shd xs) = y \<and> smap f (stl xs) = ys)"
ind_hyp_2: assumes "\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>" shows "line_integral F basis \<gamma> = (\<Sum>b\<in>basis. line_integral F {b} \<gamma>)"
elims: fixes x :: "'c exp list"   and y :: "'c exp option" assumes "ss_exps x = y" obtains "x = []" and "y = None"   | e :: "'c exp" and es :: "'c exp list" where "x = e # es" and "y = (case val_of e of None \<Rightarrow> \<lfloor>ss_exp e\<rfloor> | \<lfloor>v\<rfloor> \<Rightarrow> ss_exps es)"
interaction_bounded_by_mono: fixes "consider" :: "'i \<Rightarrow> bool"   and gpv :: "('j, 'i, 'k) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
Helly_selection: fixes f :: "nat \<Rightarrow> real \<Rightarrow> real"   and M :: "real" assumes "\<And>n x. continuous (at_right x) (f n)"   and "\<And>n. mono (f n)"   and "\<And>n x. \<bar>f n x\<bar> \<le> M" shows "\<exists>s. strict_mono s \<and> (\<exists>F. (\<forall>x. continuous (at_right x) F) \<and> mono F \<and> (\<forall>x. \<bar>F x\<bar> \<le> M) \<and> (\<forall>x. isCont F x \<longrightarrow> (\<lambda>n. f (s n) x) \<longlonglongrightarrow> F x))"
tracing_def: fixes s :: "String.literal" shows "tracing s = id"
inj_singleton: fixes A :: "'b set" shows "inj_on (\<lambda>x. {x}) A"
map_def: fixes f :: "'a" shows "\<ll>.map f = (if arr f then \<l>[cod f] \<cdot> (\<I> \<otimes> f) else null)"
G_def: shows "G = {x \<in> space M. (\<lambda>n. birkhoff_sum f n x / real n) \<longlonglongrightarrow> real_cond_exp M Invariants f x \<and> (\<lambda>n. u n x / real n) \<longlonglongrightarrow> 0}"
support_on_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.monoid_add plus zero" shows "monoid_add.support_on zero S f = {x \<in> S. f x \<noteq> zero}"
le: fixes n :: "nat" shows "integral S (\<lambda>T. \<bar>det (matrix (g' T))\<bar> * h n (g T)) \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x))"
diff_subadditive: fixes x :: "'a"   and y :: "'a" assumes "vectorspace E"   and "x \<in> E"   and "y \<in> E" shows "p (x - y) \<le> p x + p y"
trnr\<^sub>\<eta>_eq: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "src v = trg g"   and "src u = trg f"   and "\<mu> \<in> hom (v \<star> g) u" shows "trnr\<^sub>\<eta> v \<mu> = (\<mu> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, g, f] \<cdot> (v \<star> \<eta>)"
card_singleton: fixes a :: "'a" shows "card {a} = Suc 0"
le_dbm_lt: fixes a :: "'c"   and b :: "'c" assumes "Le a \<preceq> Lt b" shows "a < b"
order_less_trans: fixes x :: "'k"   and y :: "'k"   and z :: "'k" assumes "x < y"   and "y < z" shows "x < z"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegCheck: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> = set X \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G"
wf_atable_def: fixes n :: "nat"   and X :: "nat set \<times> 'b option list set" shows "wf_atable n X = (table n (fst X) (snd X) \<and> finite (fst X))"
excess_nodes_only: fixes v :: "nat" assumes "(0::'a) < excess f v" shows "v \<in> V"
simps_1: fixes x :: "'c"   and y :: "'d" shows "update x y \<langle>\<rangle> = \<langle>\<langle>\<rangle>, ((x, y), 1), \<langle>\<rangle>\<rangle>"
init_satisfies: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "\<Turnstile> (init t)"
ChiPlus_def: fixes A :: "('d, 'e, 'f) hierauto" shows "ChiPlus A \<equiv> (ChiRel A)\<^sup>+"
unity_root_mod: fixes k :: "nat"   and n :: "int" shows "unity_root k (n mod int k) = unity_root k n"
APP_def: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b" shows "f $ a \<equiv> f a"
simps_4: fixes x2 :: "'a" shows "Some x2 \<noteq> None"
bounded_a: fixes a :: "nat"   and b :: "nat list" assumes "w \<noteq> []"   and "(a, b) \<in> set (step (hd w) (ss ! hd w))" shows "a < length ss"
WTrt_binop_simps_15: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>ShiftLeft\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
nonzero_mult_divide_mult_cancel_left2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times b c) = divide a b"
difference_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CMD"   and "\<forall>w. P w z = (P w x \<and> \<not> O w y)" shows "x \<ominus> y = z"
One_not_eq_j0: shows "J.One \<noteq> J.j0"
optimize_dfn_def: fixes dnf :: "'b negation_type list list" shows "optimize_dfn dnf = map remdups (remdups dnf)"
corec2_Inl_sum: fixes s1 :: "'f \<Rightarrow> ('d, 'd JF1 + 'f, 'd JF2 + 'e) F1"   and s2 :: "'e \<Rightarrow> ('d, 'd JF1 + 'f, 'd JF2 + 'e) F2" shows "unfold2 (case_sum (F1map id Inl Inl \<circ> dtor1) s1) (case_sum (F2map id Inl Inl \<circ> dtor2) s2) \<circ> Inl = id"
set_filter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "set (filter P xs) = {x \<in> set xs. P x}"
less_induct1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "class.well_founded less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
eq_statesE: fixes s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" assumes "s \<approx> s'" obtains \<alpha> :: "'d \<Rightarrow> 'd" and \<beta> :: "'e \<Rightarrow> 'e" where "\<R>\<^sub>G \<alpha> \<beta> s = s'" and "bij \<alpha>" and "bij \<beta>"
subst_preserves_wls: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" assumes "wls s X"   and "wls (asSort ys) Y" shows "wls s (X #[Y / y]_ys)"
continuous_on_Un_local: fixes s :: "'b set"   and t :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "closedin (top_of_set (s \<union> t)) s"   and "closedin (top_of_set (s \<union> t)) t"   and "continuous_on s f"   and "continuous_on t f" shows "continuous_on (s \<union> t) f"
roots: shows "set (croots2 p) = {- b2a + csqrt bac, - b2a - csqrt bac}"
iso_tuple_UNIV_I: fixes x :: "'g" shows "x \<in> UNIV \<equiv> True"
fundfold_gallery_map: fixes s :: "'a permutation"   and Cs :: "'a set list" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "gallery Cs" shows "gallery (fst (Spair s) \<Turnstile> Cs)"
\<pi>_le_self': fixes x :: "real" assumes "1 \<le> x" shows "primes_pi x \<le> x - 1"
domainOfGraph: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "runiq (graph X f) \<and> Domain (graph X f) = X"
iso_assoc: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and h :: "('a S.E.term, 'a) S.arr" assumes "S.ide f"   and "S.ide g"   and "S.ide h"   and "S.src f = S.trg g"   and "S.src g = S.trg h" shows "S.iso (S.\<a> f g h)"
dis_rews_10: shows "is_EQ\<cdot>LT = FF"
tpart2_su: fixes V :: "Preliminaries.var set" assumes "tinfinite V" shows "tpart2 V \<subseteq> V"
dghm_inc_ObjMap_app: fixes x :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "x \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "dghm_inc \<BB> \<CC>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = x"
bij_image_Collect_eq: fixes f :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool" assumes "bij f" shows "f ` Collect P = {y. P (inv f y)}"
q63_array_def: shows "q63_array = IArray [True, True, False, False, True, False, False, True, False, True, False, False, False, False, False, False, True, False, True, False, False, False, True, False, False, True, False, False, True, False, False, False, False, False, False, False, True, True, False, False, False, False, False, True, False, False, True, False, False, True, False, False, False, False, False, False, False, False, True, False, False, False, False, False]"
is_lexer_def: fixes lexer :: "character list \<Rightarrow> nat \<Rightarrow> nat set" shows "is_lexer lexer = (\<forall>D p l. (p \<le> length D \<and> l \<in> lexer D p \<longrightarrow> p + l \<le> length D) \<and> (length D < p \<longrightarrow> lexer D p = {}))"
bind_is_OK_I2: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "h \<turnstile> ok f"   and "\<And>x h'. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> h' \<turnstile> ok g x" shows "h \<turnstile> ok Heap_Error_Monad.bind f g"
abs_triangle_ineq4: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (minus a b)) (plus (abs a) (abs b))"
IH_6: assumes "\<langle>c: t \<in> s\<rangle> \<in> set A" shows "\<langle>c: (t \<cdot> \<theta>) \<in> s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
restriction_card_mon2: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u--w \<longrightarrow> |restrict w f c| \<le> |restrict v f c|"
helpers_2: shows "inj_on p {x \<in> set zs. x < z}"
simps_5: fixes v :: "'c"   and vd :: "'c"   and ve :: "'c Tree"   and vf :: "'c Tree"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "removeLeaf (T v (T vd ve vf) (T va vb vc)) = (fst (removeLeaf (T vd ve vf)), T v (snd (removeLeaf (T vd ve vf))) (T va vb vc))"
pre_zeta_bound': fixes s :: "complex"   and a :: "real" assumes "0 < Re s"   and "0 < a" shows "cmod (pre_zeta a s) \<le> cmod s / (Re s * a powr Re s)"
prj_ssum_def: shows "PRJ('e \<oplus> 'f) = ssum_map\<cdot>PRJ('e)\<cdot>PRJ('f) oo ssum_prj"
Mapping_keys_intro: fixes f :: "('b, 'a) mapping"   and x :: "'b" assumes "Mapping.lookup f x \<noteq> None" shows "x \<in> Mapping.keys f"
finite_borel_measure_axioms: fixes k :: "nat" shows "finite_borel_measure (\<mu> k)"
one_closed: shows "\<one>\<^bsub>relative_homology_group p (discrete_topology {()}) {()}\<^esub> \<in> carrier (relative_homology_group p (discrete_topology {()}) {()})"
keys_entries: fixes k :: "'b"   and t :: "('b, 'c) RBT.rbt" shows "(k \<in> set (RBT.keys t)) = (\<exists>v. (k, v) \<in> set (RBT.entries t))"
simps_2: fixes i :: "name"   and x :: "trm"   and k :: "name" shows "subst i x (Var k) = (if i = k then x else Var k)"