hcju/isabelleps huggingface dataset:
tMap_tLast: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace" shows "tLast (tMap f t) = f (tLast t)"
cnf_of_encode_goal_state_set_ii: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "cnf ((\<Phi>\<^sub>G \<Pi>) t) = \<Union> {{{literal_formula_to_literal (encode_state_variable t (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>G) v))}} |v. v \<in> set (\<Pi>\<^sub>\<V>) \<and> (\<Pi>\<^sub>G) v \<noteq> None}"
open_contains_ball: fixes S :: "'c set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
ODNmapTr1: fixes D :: "'c Order"   and x :: "'c Order set"   and y :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D"   and "y \<in> ODnums"   and "y \<sqsubset> x" shows "y = ordinal_number (Iod D (segment D (ODNmap D y)))"
invc2_baldR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invc l"   and "invc2 r" shows "invc2 (baldR l a r)"
assms_5: shows "prv (imp \<phi>2 \<chi>)"
bnd: shows "2 * \<bar>lead_coeff f\<bar> * factor_bound f (degree_bound hs) < p ^ n"
slistmem_scons: fixes x :: "'a"   and y :: "'a"   and ys :: "[:'a:]" shows "slistmem x (y :# ys) = (x = y \<and> x \<noteq> \<bottom> \<and> ys \<noteq> \<bottom> \<or> slistmem x ys \<and> y \<noteq> \<bottom>)"
tminus_nle: fixes b :: "real"   and a :: "real" assumes "\<not> b \<le> a" shows "a \<ominus> b = 0"
convex_open_segment: fixes a :: "'b"   and b :: "'b" shows "convex (open_segment a b)"
complete_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "complete A r \<equiv> \<forall>x\<in>A. \<forall>y\<in>A - {x}. x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
interior_of_complement: fixes X :: "'b topology"   and S :: "'b set" shows "X interior_of (topspace X - S) = topspace X - X closure_of S"
Collect_eqI: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "Collect P = Collect Q"
preserves_ide: fixes a :: "'a" assumes "A.ide a" shows "A.ide (A.map a)"
c1_2: fixes x :: "real" assumes "0 \<le> x" shows "S x \<le> c1 * x"
WL_def: fixes p :: "'e process_name" shows "WL p = (\<lambda>s. W (s p) \<union> ghost_honorary_grey (s p))"
times_trilean_commutative: fixes a :: "trilean"   and b :: "trilean" shows "a \<and>? b = b \<and>? a"
holo_g: shows "g holomorphic_on {s. s0 < ereal (Re s)}"
trace_Suc: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" shows "trace n f g xs (Suc x) = the (eval (Pr n f g) (Suc x # xs)) # trace n f g xs x"
in_arcs_imp_in_arcs_ends: fixes G :: "('b, 'c) pre_digraph"   and e :: "'c" assumes "wf_digraph G"   and "e \<in> arcs G" shows "tail G e \<rightarrow>\<^bsub>G\<^esub> head G e"
ntrs_add_context_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrs fg"   and "mon_w fg e \<inter> mon_c fg cn = {}" shows "((s, c + cn), e, s', c' + cn) \<in> ntrs fg"
mod_mult_self3: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times c b) a) b = modulo a b"
is_ubD: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and u :: "'b"   and x :: "'b" assumes "class.po below"   and "po.is_ub below S u"   and "x \<in> S" shows "below x u"
diff_conv_add_uminus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus a b = plus a (uminus b)"
prems: shows "(s, (\<alpha>, t) # x # xs) \<in> seq lrs"
comp_arr_inv': fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.comp f (C.VV.inv f) = C.VV.cod f"
a_3: shows "list_all is_Check (unlabel (transaction_checks T))"
kat_1_equiv: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<cdot> y \<le> y \<cdot> d z) = (y \<cdot> a z \<le> a x \<cdot> y)"
simps: fixes a1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a2 :: "'b \<Rightarrow> bool" shows "invariant a1 a2 = (\<exists>rel I. a1 = rel \<and> a2 = I \<and> (\<forall>x xa. rel x xa \<longrightarrow> I x \<longrightarrow> I xa))"
P_Suc_0: shows "P $ Suc 0 = w0"
correctCompositionDiffLevelsS5: shows "correctCompositionDiffLevels sS5"
wls_Abs_vsubst_fresh: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "wls s X"   and "fresh xs x' X" shows "Abs xs x' (X #[x' // x]_xs) = Abs xs x X"
types_preserved_br: fixes n :: "nat"   and es0 :: "e list"   and LI :: "e list"   and vs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat"   and lholed :: "Lholed" assumes "\<lparr>[Label n es0 LI]\<rparr> \<leadsto> \<lparr>vs @ es0\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [Label n es0 LI] : ts _> ts'"   and "const_list vs"   and "length vs = n"   and "Lfilled i lholed (vs @ [$Br i]) LI" shows "\<S>\<bullet>\<C> \<turnstile> vs @ es0 : ts _> ts'"
repr_independence: fixes y :: "nat"   and H :: "nat set"   and x :: "nat" assumes "y \<in> H #>\<^bsub>G\<^esub> x"   and "x \<in> carrier G"   and "subgroup H G" shows "H #>\<^bsub>G\<^esub> x = H #>\<^bsub>G\<^esub> y"
get_lower_bound_3: fixes U :: "'a set"   and V :: "'a set" assumes "U \<in> I"   and "V \<in> I" shows "get_lower_bound U V \<subseteq> V"
Infinitesimal_FreeUltrafilterNat_iff: fixes X :: "nat \<Rightarrow> 'c" shows "(star_n X \<in> Infinitesimal) = (\<forall>u>0. \<forall>\<^sub>F n in \<U>. norm (X n) < u)"
preprm_unit_action: fixes a :: "'b"   and b :: "'b" shows "preprm_apply (preprm_unit a b) a = b"
sigma_finite_countable: shows "\<exists>A. countable A \<and> A \<subseteq> sets (M1 \<Otimes>\<^sub>M M2) \<and> \<Union> A = space (M1 \<Otimes>\<^sub>M M2) \<and> (\<forall>a\<in>A. emeasure (M1 \<Otimes>\<^sub>M M2) a \<noteq> \<infinity>)"
m'_inst_isl: fixes xs :: "'a option list"   and tstp :: "nat + nat" assumes "Mapping.lookup m' xs = Some tstp" shows "isl tstp = (0 < left I)"
simps_2: fixes a :: "('a, 'b) atm" shows "local.wtL (Neg a) = local.wtA a"
intros_7: fixes 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"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and V :: "char list"   and T :: "ty"   and e :: "expr"   and T' :: "ty" assumes "P,E,h \<turnstile> Var V :' T"   and "P,E,h \<turnstile> e :' T'"   and "P \<turnstile> T' \<le> T" shows "P,E,h \<turnstile> V:=e :' T"
pfa_insert_def: shows "pfa_insert \<equiv> \<lambda>(a, n) i x. array_shr a i 1 \<bind> (\<lambda>a'. Array.upd i x a \<bind> (\<lambda>a''. return (a'', n + (1::'c))))"
eval_Lunit: fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<l>\<^bold>[t\<^bold>]\<rbrace> = \<l>\<^sub>D[\<lbrace>Cod t\<rbrace>] \<cdot>\<^sub>D (\<I>\<^sub>D \<otimes>\<^sub>D \<lbrace>t\<rbrace>)"
filter_id_conv: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
simps_2: fixes A :: "funty"   and B :: "funty" shows "Vf (A \<sqinter> B) = Vf A @ Vf B"
suppD: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> supp f" shows "f x \<noteq> (0::'b)"
assms: shows "max_Var_floatariths fas \<le> length xs"
obtain_ideal_points: fixes H :: "circline" assumes "is_poincare_line H" obtains i1 :: "complex_homo" and i2 :: "complex_homo" where "i1 \<noteq> i2" and "ideal_points H = {i1, i2}"
not_less: fixes x :: "'b word"   and y :: "'b word" shows "(\<not> x <s y) = (y \<le>s x)"
oInv_bound1: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "normal F"   and "F 0 \<le> a" shows "F (oInv F a) \<le> a"
plain_floatarith_Sum\<^sub>e: fixes n :: "nat"   and f :: "'a \<Rightarrow> floatarith"   and xs :: "'a list" shows "plain_floatarith n (Sum\<^sub>e f xs) = list_all (\<lambda>i. plain_floatarith n (f i)) xs"
assms_2: fixes b1 :: "'a set"   and b2 :: "'a set" assumes "b1 \<in> B"   and "b2 \<in> B"   and "b1 \<inter> b2 \<noteq> {}" shows "b1 = b2"
zero_left: fixes b :: "'d" shows "0 *\<^sub>v b = (0::'c)"
simps: fixes \<pi>l :: "('state \<times> 'aprop set) stream list" shows "pointerOf \<pi>l = (if \<pi>l \<noteq> [] then stateOf (last \<pi>l) else s0)"
step_2: fixes gpv :: "('a, 'call, 'ret) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "expectation_gpv' gpv \<le> expectation_gpv2 (\<lambda>a. case a of (x, s) \<Rightarrow> f x) (inline callee gpv s)"
le_funI2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>i j. f i j \<le> g i j" shows "f \<le> g"
endofunctor_axioms: shows "endofunctor (\<cdot>\<^sub>D) src\<^sub>D"
sub1: shows "(\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO ODE1}. FVT (\<sigma> y)) \<subseteq> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO (OProd ODE1 ODE2)}. FVT (\<sigma> y))"
SPEC_nofail: fixes X :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "X \<le> SPEC \<Phi>" shows "nofail X"
finprod_closed: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'e \<Rightarrow> 'c"   and A :: "'e set" assumes "Group.comm_monoid G"   and "f \<in> A \<rightarrow> carrier G" shows "finprod G f A \<in> carrier G"
valid_mmsaux_tuple_in_keys: fixes args :: "Monitor.args"   and cur :: "nat"   and nt :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'a option list set) queue"   and data_in :: "(nat \<times> 'a option list set) queue"   and tuple_in :: "('a option list, nat) mapping"   and tuple_since :: "('a option list, nat) mapping"   and ys :: "(nat \<times> 'a option list set) list" assumes "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) ys" shows "Mapping.keys tuple_in = snd ` {tas \<in> ts_tuple_rel (set (linearize data_in)). valid_tuple tuple_since tas}"
existence_of_het0: fixes p :: "nat"   and x :: "real list" assumes "p = het x \<and> 0 < p \<and> pos x" shows "\<exists>y. gmean x < gmean y \<and> \<gamma>_eq (x, y) \<and> het y = 0 \<and> pos y"
minus_scalar_prod_distrib: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec"   and v\<^sub>3 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "(v\<^sub>1 - v\<^sub>2) \<bullet> v\<^sub>3 = v\<^sub>1 \<bullet> v\<^sub>3 - v\<^sub>2 \<bullet> v\<^sub>3"
that: fixes s :: "nat" assumes "start = Index s"   and "s < IArray.length ss" shows "thesis"
bigo_const_mult3: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "f \<in> O(\<lambda>x. c * f x)"
borel_measurable_implies_simple_function_sequence: fixes u :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "u \<in> borel_measurable M" shows "\<exists>f. incseq f \<and> (\<forall>i. (\<forall>x. f i x < top) \<and> simple_function M (f i)) \<and> u = Sup (range f)"
antisym_conv: fixes y :: "'al"   and x :: "'al" assumes "y \<in> U"   and "x \<in> U"   and "x \<le>\<^sub>o\<^sub>w y" shows "(y \<le>\<^sub>o\<^sub>w x) = (x = y)"
left2right: fixes z :: "hf" assumes "z \<^bold>\<in> s" shows "z \<^bold>\<in> insf s k y"
N: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap" shows "N \<in> F"
hom_dvd: fixes p :: "'a"   and q :: "'a" assumes "p dvd q" shows "[:p:] dvd [:q:]"
assms_2: shows "ptr \<noteq> cast new_shadow_root_ptr"
comp_assoc: fixes h :: "'b"   and g :: "'b"   and f :: "'b" shows "(h \<cdot>\<^sub>B g) \<cdot>\<^sub>B f = h \<cdot>\<^sub>B g \<cdot>\<^sub>B f"
rrsmult_distrib_right: fixes f :: "'r \<Rightarrow> 'a"   and r :: "'r"   and s :: "'r" assumes "\<And>x y. \<lbrakk>x \<in> R; y \<in> R\<rbrakk> \<Longrightarrow> f (x + y) = f x + f y"   and "r \<in> R"   and "s \<in> R" shows "(r + s) \<currency> f = r \<currency> f + s \<currency> f"
set_less_eq_aux''_finite: assumes "finite UNIV" shows "(\<sqsubseteq>''') = (\<sqsubseteq>')"
main_part: fixes S :: "'c set"   and M :: "'c measure" assumes "S \<in> sets M" shows "main_part M S = S"
ih: shows "apps s (t' # us) >\<^sub>t apps s (t # us)"
eval_list_disj: fixes F :: "atom fm list"   and L :: "real list" shows "eval (list_disj F) L = (\<exists>f\<in>set F. eval f L)"
Mul_interfree_Mutator_Collector: fixes j :: "nat"   and n :: "nat" assumes "j < n" shows "interfree_aux (Some (Mul_Mutator j n), {}, Some (Mul_Collector n))"
aux_2_def: fixes g :: "'g"   and n :: "'node" shows "aux_2 g n = foldr (aux_1 g n) (sorted_list_of_set (uses g n))"
le_add_same_cancel2: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b + a) = ((0::'a) \<le> b)"
lexord_take_index_conv: fixes x :: "'b list"   and y :: "'b list"   and r :: "('b \<times> 'b) set" shows "((x, y) \<in> lexord r) = (\<^bold>|x\<^bold>| < \<^bold>|y\<^bold>| \<and> take \<^bold>|x\<^bold>| y = x \<or> (\<exists>i<min \<^bold>|x\<^bold>| \<^bold>|y\<^bold>|. take i x = take i y \<and> (x ! i, y ! i) \<in> r))"
orderI: fixes a :: "nat"   and b :: "nat" assumes "a = gcd a b" shows "a dvd b"
subtract_mask_2: fixes p :: "'b word"   and n :: "nat" shows "p - (p AND NOT (mask n)) = p AND mask n"
order_trans_rules_21: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y < z" shows "x < z"
L_def: shows "L \<equiv> (ideal F \<div> f) \<inter> P[X]"
upper: fixes x :: "'a option" assumes "x \<in> Y" shows "ord_option ord x y"
combine_nrows: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix"   and q :: "nat"   and B :: "'d matrix" assumes "f (0::'c) (0::'d) = (0::'b)"   and "nrows A \<le> q"   and "nrows B \<le> q" shows "nrows (combine_matrix f A B) \<le> q"
sum_is_minimal: fixes N1 :: "'c set"   and N2 :: "'c set"   and N :: "'c set" assumes "LinearCombinations.submodule R N1 M"   and "LinearCombinations.submodule R N2 M"   and "LinearCombinations.submodule R N M" shows "(submodule_sum N1 N2 \<subseteq> N) = (N1 \<subseteq> N \<and> N2 \<subseteq> N)"
enat_coinduct: fixes P :: "enat \<Rightarrow> enat \<Rightarrow> bool"   and m :: "enat"   and n :: "enat" assumes "P m n"   and "\<And>m n. P m n \<Longrightarrow> (m = 0) = (n = 0) \<and> (m \<noteq> 0 \<longrightarrow> n \<noteq> 0 \<longrightarrow> P (epred m) (epred n) \<or> epred m = epred n)" shows "m = n"
gpath_is_empty_impl_def: fixes s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gpath_is_empty_impl s = path_is_empty_impl (snd s)"
sorted_tl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (tl xs)"
cxt_inv: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and E :: "('a, 'c, 'd) Stmt list"   and c :: "('a, 'c, 'd) Stmt"   and c' :: "('a, 'c, 'd) Stmt" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "sifum_lang_no_dma.cxt_to_stmt E c = c'"   and "\<And>p q. c' \<noteq> p ;; q" shows "E = [] \<and> c' = c"
sel_2: fixes x1 :: "'c"   and x2 :: "'c stream" shows "stl (x1 ## x2) = x2"
almost_full_onI: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>f. \<forall>i. f i \<in> A \<Longrightarrow> good P f" shows "almost_full_on P A"
inter_sorted_correct_col: fixes xs :: "'b list"   and ys :: "'b list" assumes "Sorted_Less.sorted (rev xs)"   and "Sorted_Less.sorted (rev ys)" shows "inter_sorted_rev xs ys = rev (sorted_list_of_set (set xs \<inter> set ys))"
a_get_mode_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "get_mode shadow_root_ptr = get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode"
commute: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a"   and b :: "'a" assumes "merkle_interface h bo m" shows "m a b = m b a"
rel_spmf_mono: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'g spmf"   and g :: "'h spmf"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_spmf B f g"
IH_1: fixes x :: "perm" assumes "x \<bullet> t1 =\<^sub>\<alpha> t2" shows "L_transform_Tree (x \<bullet> t1) = L_transform_Tree t2"
semiring_normalization_rules_29: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times x x = power.power one times x 2"
nsD': fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>ns v" shows "u \<noteq> \<epsilon>"
A4b: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<^bold>\<not>\<P> x \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<not>\<P> x) w)\<rfloor>"
le: fixes z :: "complex" assumes "z \<in> ball a r" shows "cmod (deriv f z) \<le> 2 * cmod (deriv f a)"
prems_5: shows "ps (S t i) p \<noteq> None"
\<HH>_def': shows "\<HH> = \<GG> \<circ>\<^sub>A\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<FF>"
swap_apply_1: fixes a :: "'e"   and b :: "'e"   and f :: "'e \<Rightarrow> 'd" shows "Fun.swap a b f a = f b"
func_pre: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and A :: "'e set" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> A" shows "f \<in> {j. j \<le> n} \<rightarrow> A"
replace2: fixes f' :: "'a"   and ls :: "'a list"   and fs :: "'a list" assumes "f' \<notin> set ls" shows "replace f' fs ls = ls"
normal_form_with_flatten_sigma_2_length_def: fixes \<phi> :: "'b ltln" shows "normal_form_with_flatten_sigma_2_length \<phi> \<equiv> \<Sum>(M, N)\<in>{(M, N) |M N. M \<subseteq> subformulas\<^sub>\<mu> \<phi> \<and> N \<subseteq> subformulas\<^sub>\<nu> \<phi>}. normal_form_disjunct_with_flatten_sigma_2_length \<phi> M N"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<^bold>\<squnion> b) \<^bold>\<squnion> c = a \<^bold>\<squnion> b \<^bold>\<squnion> c"
intros: fixes S :: "'procS" shows "SourceTerm S \<R>\<lbrakk>\<cdot>\<rbrakk>R TargetTerm (\<lbrakk>S\<rbrakk>)"
many_strongly_connected_iff_7: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.relation_algebra_rtc minus uminus inf less_eq less sup bot top composition converse unit star" shows "(star x = star (converse x)) = (composition (converse x) (star (converse x)) = composition x (star x))"
Quotient3_abs_rep: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g"   and a :: "'h" assumes "Quotient3 R Abs Rep" shows "Abs (Rep a) = a"
Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>"   and \<alpha> :: "'c" assumes "Formula.hereditarily_fs t\<^sub>\<alpha>" shows "Formula.hereditarily_fs (Formula.Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)"
INF_nat_shift: fixes f :: "nat \<Rightarrow> 'b::Inf" shows "\<Sqinter> ((f::nat \<Rightarrow> 'b::Inf) ` {i::nat. 0 < i}) = (\<Sqinter>i::nat. f (Suc i))"
upper_bound_C: shows "\<exists>C\<ge>1. \<forall>n. B (Suc n) \<le> C * Max {B i |i. i \<le> n}"
le_sup_iff: fixes x :: "'d multiset"   and y :: "'d multiset"   and z :: "'d multiset" shows "(x \<union># y \<subseteq># z) = (x \<subseteq># z \<and> y \<subseteq># z)"
simplex_retraction2: fixes x :: "'a set" assumes "x \<in> g \<turnstile> X" shows "g ` x = x"
function_ring_car_memI: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'a" assumes "\<And>a. a \<in> S \<Longrightarrow> f a \<in> carrier F"   and "\<And>a. a \<notin> S \<Longrightarrow> f a = undefined" shows "f \<in> carrier (function_ring S F)"
has_subprob_density_distr_Fst: fixes M :: "val measure"   and t1 :: "pdf_type"   and t2 :: "pdf_type"   and f :: "val \<Rightarrow> ennreal" assumes "has_subprob_density M (stock_measure (PRODUCT t1 t2)) f" shows "has_subprob_density (distr M (stock_measure t1) (op_sem Fst)) (stock_measure t1) (\<lambda>x. \<integral>\<^sup>+ y. f <|x, y|> \<partial>stock_measure t2)"
insert_compare_element: fixes a :: "'b"   and A :: "'b set"   and b :: "'b" assumes "a \<notin> A"   and "insert b A = insert a A" shows "b = a"
product_complement_in_complements_sum: fixes x :: "i"   and y :: "i" assumes "CEMC"   and "x \<noteq> u"   and "y \<noteq> u" shows "P (\<midarrow> (x \<otimes> y)) (\<midarrow> x \<oplus> \<midarrow> y)"