hcju/isabelleps huggingface dataset:
sup_state_opt_Some_Some: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list" shows "P \<turnstile> \<lfloor>a\<rfloor> \<le>' \<lfloor>b\<rfloor> = P \<turnstile> a \<le>\<^sub>i b"
A_def: shows "A = (SIGMA n:{1..6}. {T' n<..2 * T' n + 1})"
VDMAx: fixes A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "A \<in> G" shows "G \<rhd> Call : A"
address_of_simps_1: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "address_of (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = {a}"
less_diff_le_imp_less: fixes i :: "nat"   and j :: "nat"   and m :: "nat"   and n :: "nat" assumes "i < j - m"   and "n \<le> m" shows "i < j - n"
blinding_of_G_root_hash: fixes boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rha :: "'e \<Rightarrow> 'f"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and rhb :: "'g \<Rightarrow> 'h" assumes "boa \<le> vimage2p rha rha (=)"   and "bob \<le> vimage2p rhb rhb (=)" shows "blinding_of_G rha boa rhb bob \<le> vimage2p (root_hash_G rha rhb) (root_hash_G rha rhb) (=)"
neighborhoods_def: fixes x :: "'d" shows "im0.target.neighborhoods x \<equiv> {U. U = {} \<and> x \<in> U}"
diff_single_eq_union: fixes x :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" assumes "x \<in># M" shows "(remove1_mset x M = N) = (M = add_mset x N)"
assms: shows "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
of_int_eq_1_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = one) = (z = 1)"
id_arrow: fixes A :: "'b" assumes "A \<in> Ob" shows "Id A \<in> Ar"
red_in_Hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a" shows "monoidal_language.red C a \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = a \<and> monoidal_language.Cod C t = monoidal_language.Diagonalize C a}"
option_rel_Some1: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d option" shows "rel_option A (Some x) y = (\<exists>y'. y = Some y' \<and> A x y')"
set_bits_int_def: fixes f :: "nat \<Rightarrow> bool" shows "set_bits f = (if \<exists>n. \<forall>m\<ge>n. f m = f n then let n = LEAST n. \<forall>m\<ge>n. f m = f n in signed_take_bit n (horner_sum of_bool 2 (map f [0..<Suc n])) else 0)"
homotopic_paths_def: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" shows "homotopic_paths s p q \<equiv> homotopic_with_canon (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
a: shows "R[a\<turnstile>c>b] = OrR2 <c>.M e"
perm_append_Cons: fixes a :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "a # xs @ ys <~~> xs @ a # ys"
r_ad: fixes p :: "'a" shows "r (ad p) = ad p"
simps_1: fixes ys :: "'a list" shows "local.filter_min_aux [] ys = ys"
les_def: fixes s :: "'s"   and s' :: "'s" shows "les s s' \<equiv> \<forall>ni n. (ni, n) \<in> R s \<longrightarrow> (ni, n) \<in> R s'"
fBexI: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and A :: "'c fset" assumes "P x"   and "x |\<in>| A" shows "fBex A P"
sees_field_i_i_i_o_o_iI: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "String.literal"   and xc :: "ty"   and xd :: "fmod"   and xe :: "String.literal" assumes "x \<turnstile> xa sees xb:xc (xd) in xe" shows "pred.eval (sees_field_i_i_i_o_o_i x xa xb xe) (xc, xd)"
unat_ucast_less_no_overflow: fixes n :: "nat"   and f :: "'b word" assumes "n < 2 ^ LENGTH('b)"   and "unat f < n" shows "f < word_of_nat n"
lookup_map_of: fixes t :: "(('c \<times> 'd) \<times> 'e) tree"   and x :: "'c" assumes "sorted1 (Tree2.inorder t)" shows "lookup t x = AList_Upd_Del.map_of (Tree2.inorder t) x"
fin: shows "finite ({i. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i} \<union> {k})"
simps: fixes pi :: "('a \<times> 'a) list" shows "pi \<bullet> () = ()"
uminus_l_inv_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "- A + A = 0\<^sub>m nr nc"
ND_Rules_Inst_axioms: shows "ND_Rules_Inst freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"
restrict_map_id: fixes m :: "'b \<Rightarrow> 'c option" shows "m |` dom m = m"
exists_diff: fixes P :: "'a set \<Rightarrow> bool" shows "(\<exists>S. P (- S)) = (\<exists>S. P S)"
ideal_prod_eq_Inter_aux: fixes I :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "I \<in> {..Suc n} \<rightarrow> {J. ideal J R}"   and "\<And>i j. \<lbrakk>i \<le> Suc n; j \<le> Suc n; i \<noteq> j\<rbrakk> \<Longrightarrow> I i <+> I j = carrier R" shows "finprod (ideals_set R) I {..n} <+> I (Suc n) = carrier R"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "a \<rightarrow> b" shows "b \<in> A"
ardenable_set_union: fixes rhs :: "'b trm set"   and rhs' :: "'b trm set" assumes "ardenable rhs"   and "ardenable rhs'" shows "ardenable (rhs \<union> rhs')"
inj_term_simps_1: fixes c :: "stmt" shows "\<langle>c\<rangle> = In1r c"
WTrt_elim_cases_18: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and o' :: "'addr expr"   and e :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> sync(o') e : T" obtains T :: "ty" where "P,E,h \<turnstile> o' : T" and "is_refT T" and "P,E,h \<turnstile> e : Ta__"
apply_ncrit_superset: fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and qs :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "set (apply_ncrit crit data gs bs hs ps) \<subseteq> set (apply_ncrit crit data gs bs hs (ps @ qs))"
is_Skip_def: fixes step :: "('e, 'f) step" shows "is_Skip step = (\<exists>x2. step = Skip x2)"
aform_val_def: fixes e :: "nat \<Rightarrow> real"   and X :: "'a \<times> 'a pdevs" shows "aform_val e X = fst X + pdevs_val e (snd X)"
sum_list_sets_double: fixes A :: "'a"   and B :: "'a" shows "(\<Sum>X\<leftarrow>[A, B]. X) = A + B"
progress_unop_f: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and cs :: "e list"   and t :: "t"   and e :: "b_e"   and iop :: "unop_f"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [t]"   and "is_float_t t"   and "const_list cs"   and "e = Unop_f t iop" shows "\<exists>a s' vs' es'. \<lparr>s;vs;cs @ [$e]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
nonequiv_Ssegment: fixes D :: "'c Order"   and a :: "'c" assumes "Worder D"   and "a \<in> carrier D" shows "\<not> ord_equiv D (SIod D (Ssegment D a))"
MP: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (F::'b formula)"   and "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<rightarrow> (G::'b formula)" shows "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (G::'b formula)"
FAccRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s,b\<rangle> \<rightarrow> \<langle>e'\<bullet>F{D},s',b'\<rangle>"
Abs_formula_inverse: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" assumes "Formula.hereditarily_fs t\<^sub>\<alpha>" shows "Formula.formula.Rep_formula (Formula.formula.Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
cfun_below_ID: fixes f :: "'a \<rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "f \<sqsubseteq> ID"   and "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> y"
h1: shows "higherPowers i j ` ({lowerPowers i j m} \<inter> {x. \<forall>j\<in>{i..<i + j}. lookup x j = 0}) = {m}"
distinct_1: fixes x21 :: "'c"   and x22 :: "'c llist" shows "LNil \<noteq> LCons x21 x22"
resource_of_oracle_parametric: shows "rel_fun (rel_fun (BNF_Def.Grp UNIV id) (rel_fun (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (rel_spmf (rel_prod (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV id))))) (rel_fun (BNF_Def.Grp UNIV id) (rel_resource (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g))) resource_of_oracle resource_of_oracle"
rreq_rrep_sn_def: fixes m :: "msg" shows "rreq_rrep_sn m \<equiv> case m of Rreq x xa xb xc xd osnc xe xf \<Rightarrow> 1 \<le> osnc | Rrep x xa dsnc xb xc \<Rightarrow> 1 \<le> dsnc | _ \<Rightarrow> True"
closure_sound: fixes as :: "nat list"   and r :: "nat rexp"   and s :: "nat rexp"   and ps :: "(nat rexp \<times> nat rexp) list" assumes "closure as ([(r, s)], []) = Some ([], ps)"   and "atoms r \<union> atoms s \<subseteq> set as" shows "lang (set as) r = lang (set as) s"
Cprems_2: shows "(a # al) @ e # bl = al' @ e # bl'"
isin: fixes empty :: "'b"   and insert :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and delete :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and isin :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and inorder :: "'b \<Rightarrow> 'c list"   and inv :: "'b \<Rightarrow> bool"   and t :: "'b"   and x :: "'c" assumes "Set_by_Ordered empty insert delete isin inorder inv"   and "inv t \<and> Sorted_Less.sorted (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
mono: fixes m :: "'a"   and n :: "'a" assumes "m \<le> n" shows "F n \<subseteq> F m"
157: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> y) \<sqinter> - x = (top \<squnion> y) \<sqinter> - x"
AE_T': fixes P :: "'s stream \<Rightarrow> bool"   and I :: "'s pmf" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "almost_everywhere (K.T' I) P = (\<forall>s\<in>set_pmf I. AE x in K.T s. P (s ## x))"
assms_2: shows "set_pmf p \<inter> A \<noteq> {}"
cut_less_Min_empty: fixes t :: "'b"   and I :: "'b set" assumes "t \<le> iMin I" shows "I \<down>< t = {}"
gt_trans: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and u :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "rpo_optim ground_heads_var gt_sym extf arity_sym arity_var"   and "rpo_optim.gt ground_heads_var gt_sym extf u t"   and "rpo_optim.gt ground_heads_var gt_sym extf t s" shows "rpo_optim.gt ground_heads_var gt_sym extf u s"
intro: fixes Prec_F_g :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>g. minimal_element (Prec_F_g g) UNIV" shows "tiebreaker_lifting_axioms Prec_F_g"
assms_2: shows "h \<turnstile> assigned_nodes slot' \<rightarrow>\<^sub>r nodes'"
SE_rel: fixes rv1 :: "'a \<times> nat"   and res :: "('a \<times> nat) edge list"   and rv2 :: "'a \<times> nat"   and c :: "('b, 'c) conf" assumes "RedBlack prb"   and "SubRel.subpath (red prb) rv1 res rv2 (subs prb)"   and "se_star (confs prb rv1) (trace (ui_es res) (labelling (black prb))) c" shows "c \<sqsubseteq> confs prb rv2"
multiplicity_decompose: fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p" shows "\<not> p dvd x div p ^ multiplicity p x"
compl_struct_f4_red: shows "compl_struct f4_red"
v_max: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G"   and "g \<notin> M"   and "q g \<odot> g \<noteq> 0" shows "lt (q g \<odot> g) \<prec>\<^sub>t v"
rw: shows "- interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ ia $ j = - mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j) $ ia $ j"
list_before_irrefl_eq_distinct: fixes l :: "'b list" shows "irrefl (list_before_rel l) = distinct l"
indRelLSTPO_refl: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "refl SRel"   and "refl TRel" shows "refl (indRelLSTPO SRel TRel)"
cut_le_finite: fixes I :: "'a set"   and t :: "'a" assumes "finite I" shows "finite (I \<down>\<le> t)"
1: fixes s' :: "'a" assumes "s' \<in> ss" shows "\<exists>rs. s' = s0 \<star> rs \<and> set rs \<subseteq> rset"
approx_mult_SReal1: fixes a :: "real star"   and x :: "real star" assumes "a \<in> \<real>"   and "x \<approx> 0" shows "x * a \<approx> 0"
ffield_pair: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r" obtains b :: "V" where "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<or> [b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
dvd_div_mult_self: fixes a :: "'b"   and b :: "'b" assumes "a dvd b" shows "b div a * a = b"
spr_tObsC_trc_aux: fixes t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace" assumes "(t, t') \<in> (\<Union> (range (relations SPR.MC)))\<^sup>*" shows "tObsC t = tObsC t'"
prod_mset_prime_factorization_int: fixes n :: "int" assumes "0 < n" shows "\<Prod>\<^sub># (prime_factorization n) = n"
Reals_0: shows "(0::'b) \<in> \<real>"
\<alpha>_simp: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and h :: "'c local.term set" assumes "arr f"   and "arr g"   and "arr h" shows "assoc f g h = tensor f (tensor g h) \<cdot> assoc (local.dom f) (local.dom g) (local.dom h)"
simps: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" shows "mgu s t = (case unify [(s, t)] [] of None \<Rightarrow> None | Some res \<Rightarrow> Some (subst_of res))"
rev_isinceI: fixes t :: "nat"   and I :: "nat set"   and Q :: "nat \<Rightarrow> bool"   and P :: "nat \<Rightarrow> bool" assumes "t \<in> I"   and "Q t"   and "\<And>t'. t' \<in> I \<down>> t \<Longrightarrow> P t'" shows "P t'. t' \<S> t I. Q t"
univ_ce_is_not_comp3: shows "\<not> computable univ_ce"
canonical_basis_ell2_def: shows "canonical_basis = map ket enum_class.enum"
prob_space_distr: fixes f :: "'b \<times> 'c \<Rightarrow> 'd"   and M' :: "'d measure" assumes "Pxy.random_variable M' f" shows "prob_space (distr (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) M' f)"
FrameStatEqImpCompose: fixes F :: "'b frame"   and G :: "'b frame"   and H :: "'b frame"   and I :: "'b frame" assumes "F \<simeq>\<^sub>F G"   and "G \<hookrightarrow>\<^sub>F H"   and "H \<simeq>\<^sub>F I" shows "F \<hookrightarrow>\<^sub>F I"
dom_def: fixes m :: "'d \<Rightarrow> 'e option" shows "dom m = {a. m a \<noteq> None}"
relax_outgoing'_def: fixes u :: "'v"   and du :: "nat"   and V :: "'v \<Rightarrow> 'a option"   and Q :: "'v \<Rightarrow> nat option"   and v :: "'v" shows "relax_outgoing' u du V Q v \<equiv> case w (u, v) of enat d \<Rightarrow> case Q v of None \<Rightarrow> if v \<in> dom V then None else Some (du + d) | Some d' \<Rightarrow> Some (min d' (du + d)) | \<infinity> \<Rightarrow> Q v"
d_llp1: fixes x :: "'a"   and y :: "'a" assumes "d x \<le> d y" shows "x \<le> d y \<cdot> x"
mat_base_case_R: shows "M_mat_R [[1], [0], [- 1]] [([], []), ([0], []), ([], [0])] = mat_of_rows_list 3 [[1, 1, 1], [0, 1, 0], [1, 0, - 1]]"
distinct_5: fixes x1 :: "'b"   and x41 :: "'b exp"   and x42 :: "'b exp" shows "Var x1 \<noteq> Div x41 x42"
Term_Var: fixes i :: "name" shows "Term (q_Var i)"
assms_4: shows "x \<notin> Fvars \<chi>"
ffilter_finsert: fixes f :: "'a \<Rightarrow> bool"   and a :: "'a"   and s :: "'a fset" shows "ffilter f (finsert a s) = (if f a then finsert a (ffilter f s) else ffilter f s)"
annul_mem: fixes asi :: "word8"   and addr :: "word32"   and s :: "'b sparc_state"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32"   and new_state_var :: "sparc_state_var" shows "fst (memory_read asi addr s) = fst (memory_read asi addr (s\<lparr>state_var := new_state_var, cpu_reg := new_cpu_reg\<rparr>))"
1: shows "local.map (g, a) = S.mkArr (local.set (C.cod g, a)) (local.set (C.dom g, a)) (\<phi> (C.dom g, a) \<circ> (\<lambda>h. a \<cdot> h \<cdot> g) \<circ> \<psi> (C.cod g, a))"
A: shows "(rGamma has_field_derivative (- 1) ^ n * fact n) (at (- of_nat (Suc n) + 1))"
sorted_asc: fixes j :: "nat"   and i :: "nat"   and ss :: "'b list"   and k :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd" assumes "j \<le> i"   and "i < length ss"   and "\<forall>x\<in>set ss. \<forall>y\<in>set ss. k x \<le> k y \<longrightarrow> f y \<le> f x"   and "sorted (map k ss)" shows "f (ss ! i) \<le> f (ss ! j)"
lcm_mult_unit2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "lcm b (c * a) = lcm b c"
D'_def: shows "D' = (\<lambda>n. complex_of_real (a + real n) powr - s)"
set_membership_equality_technicalD: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" assumes "{x} \<union> set xs = {y} \<union> set ys" shows "x = y \<or> y \<in> set xs"
cong_def: fixes H :: "'a set" shows "cong H = (\<lambda>x y. cong_l H x y \<and> cong_r H x y)"
assms_14: fixes \<phi> :: "'a formula"   and \<psi> :: "'a formula"   and I :: "\<I>" assumes "fv (Neg \<phi>) \<subseteq> fv \<psi>"   and "\<not> safe_formula (Neg \<phi>)"   and "P \<phi>"   and "P \<psi>" shows "P (Since (Neg \<phi>) I \<psi>)"
succs: fixes g' :: "'a"   and succs :: "'a \<Rightarrow> 'a list"   and g :: "'a"   and g'' :: "'a" assumes "g' \<in> set (succs g)"   and "RTranClp succs g' g''" shows "RTranClp succs g g''"
times_mk_dimvec: fixes xs :: "'a list"   and ys :: "'a list" shows "mk_dimvec xs \<cdot> mk_dimvec ys = (if length xs = CARD('b) \<and> length ys = CARD('b) then mk_dimvec (map2 (+) xs ys) else if length xs = CARD('b) then mk_dimvec xs else mk_dimvec ys)"
disjoint_def: fixes x :: "'a list"   and y :: "'a list" shows "disjoint x y \<equiv> set x \<inter> set y = {}"
blinfun_apply_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. bounded_linear f}" obtains x :: "'a \<Rightarrow>\<^sub>L 'b" where "y = blinfun_apply x"
mod_neq_imp_diff_not_dvd: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a mod m \<noteq> b mod m"   and "a \<le> b" shows "\<not> m dvd b - a"
mult2_alt_ns_def: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "mult2_alt_ns ns s = {(X, Y). \<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s))}"
of_nat_eq: fixes k :: "nat" shows "of_nat k = Abs_bit1 (int k mod int CARD('b bit1))"
le: fixes numnoabs :: "Ferrack.num \<Rightarrow> bool"   and t :: "Ferrack.num"   and bs :: "real list" assumes "numnoabs t" shows "Ifm bs (case rsplit0 t of (x, xa) \<Rightarrow> le x xa) = Ifm bs (Le t) \<and> isrlfm (case rsplit0 t of (x, xa) \<Rightarrow> le x xa)"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (f \<otimes> \<I>)"
hom_in_carrier: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme"   and x :: "'b" assumes "h \<in> hom G H"   and "x \<in> carrier G" shows "h x \<in> carrier H"
zero_eq_1_divide_iff: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(zero = divide one a) = (a = zero)"
normal_simps_4: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "normal (And \<phi>\<^sub>1 \<phi>\<^sub>2) = (normal \<phi>\<^sub>1 \<and> normal \<phi>\<^sub>2)"
rep_eq: fixes x :: "'a perm"   and xa :: "'a perm" shows "(\<langle>$\<rangle>) (x * xa) = (\<langle>$\<rangle>) x \<circ> (\<langle>$\<rangle>) xa"
refine: fixes ai :: "'b list"   and a :: "'c set"   and sctni :: "real list sctn"   and sctn :: "'c sctn" assumes "(ai, a) \<in> appr_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(RETURN (below_sctn_impl ai sctni), below_sctn $ a $ sctn) \<in> \<langle>bool_rel\<rangle>nres_rel"
gram_schmidt0_result_1: fixes ws :: "complex vec list" assumes "set ws \<subseteq> carrier_vec n" shows "set (gram_schmidt0 n ws) \<subseteq> carrier_vec n"
count_of_multiset: fixes xs :: "('b \<times> nat) list" shows "count_of xs \<in> multiset"
that: fixes d' :: "'a" assumes "u(c := d') \<in> [M]\<^bsub>v,n\<^esub>" shows "thesis"
compl_le_swap1: fixes y :: "'b upred"   and x :: "'b upred" assumes "(\<not> x) \<sqsubseteq> y" shows "(\<not> y) \<sqsubseteq> x"
phis_disj_2: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list"   and n' :: "'node"   and vs' :: "'val list" assumes "phis g (n, v) = Some vs"   and "phis g (n', v) = Some vs'" shows "vs = vs'"
l_get_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_mode get_mode_locs type_wf \<equiv> type_wf = ShadowRootClass.type_wf \<and> get_mode = get_mode \<and> get_mode_locs = get_mode_locs"
finite_imp_nat_seg_image_inj_on: fixes A :: "'g set" assumes "finite A" shows "\<exists>n f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
intros: fixes a :: "'a"   and x :: "'b" shows "[a].x \<in> ABS_set"
mset_le_decr_left_3: fixes c :: "'c"   and a :: "'c multiset"   and b :: "'c multiset" assumes "add_mset c a \<subseteq># b" shows "a \<subseteq># b"
bij_pointE: fixes f :: "'q \<Rightarrow> 'r"   and y :: "'r" assumes "bij f" obtains x :: "'q" where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
assms: shows "f \<equiv> ASSERT \<Phi> \<bind> (\<lambda>_. RETURN r)"
Diff_infinite_finite: fixes T :: "'b set"   and S :: "'b set" assumes "finite T"   and "infinite S" shows "infinite (S - T)"