hcju/isabelleps huggingface dataset:
two_reduced_heads_imp_reduced_alt': fixes s :: "'w"   and t :: "'w"   and w :: "'w"   and as :: "'w list"   and bs :: "'w list"   and n :: "nat" assumes "s \<noteq> t"   and "S_reduced_for w (s # as)"   and "S_reduced_for w (t # bs)"   and "order.greater_eq (add_order (s + t)) n \<or> add_order (s + t) = 0" shows "\<exists>cs. S_reduced_for w (alternating_list n s t @ cs) \<or> S_reduced_for w (alternating_list n t s @ cs)"
simps_12: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'a"   and x3 :: "nat" shows "(case Rg x3 of Ip x \<Rightarrow> f1 x | Op x \<Rightarrow> f2 x | Rg x \<Rightarrow> f3 x) = f3 x3"
cases: fixes r :: "('b, 'c) pre_digraph" obtains verts :: "'b set" and arcs :: "'c set" and tail :: "'c \<Rightarrow> 'b" and head :: "'c \<Rightarrow> 'b" where "r = \<lparr>verts = verts, arcs = arcs, tail = tail, head = head\<rparr>"
s'_a_equals_s_a: fixes A :: "type_scheme list"   and t :: "'a"   and S :: "nat \<Rightarrow> typ" shows "$ (\<lambda>n. if n \<in> free_tv A \<union> free_tv t then S n else TVar n) A = $ S A"
prime_1_mod_4_sum_of_squares_unique: fixes p :: "nat" assumes "prime p"   and "[p = 1] (mod 4)" shows "\<exists>!(x, y). x \<le> y \<and> x\<^sup>2 + y\<^sup>2 = p"
open_inter_closure_rel_interior: fixes S :: "'a set"   and A :: "'a set" assumes "convex S"   and "open A" shows "(A \<inter> closure S = {}) = (A \<inter> rel_interior S = {})"
farkas_coefficients_def: fixes cs :: "constraint set"   and C :: "(rat \<times> constraint) list" shows "farkas_coefficients cs C = (\<exists>d rel. (\<forall>(r, c)\<in>set C. c \<in> cs \<and> is_le (r *R c) \<and> r \<noteq> 0) \<and> (\<Sum>(r, c)\<leftarrow>C. lec_of_constraint (r *R c)) = Le_Constraint rel 0 d \<and> (rel = Leq_Rel \<and> d < 0 \<or> rel = Lt_Rel \<and> d \<le> 0))"
dg_GRPH_ObjD: fixes \<AA> :: "V"   and \<alpha> :: "V" assumes "\<AA> \<in>\<^sub>\<circ> dg_GRPH \<alpha>\<lparr>Obj\<rparr>" shows "digraph \<alpha> \<AA>"
app_vrrestrictionI: fixes b :: "V"   and A :: "V"   and a :: "V"   and r :: "V" assumes "b \<in>\<^sub>\<circ> A"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A"
Collect_mono_iff: fixes P :: "'h \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"
imult_is_0: fixes m :: "enat"   and n :: "enat" shows "(m * n = 0) = (m = 0 \<or> n = 0)"
assms: shows "swapidseq_ext A (Suc n) p"
hsize_apps: fixes t :: "('c, 'd) tm"   and ts :: "('c, 'd) tm list" shows "hsize (apps t ts) = hsize t + sum_list (map hsize ts)"
mult_le_cancel_right2: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (times a c) c = ((less zero c \<longrightarrow> less_eq a one) \<and> (less c zero \<longrightarrow> less_eq one a))"
list_emb: fixes f :: "'b"   and m :: "nat"   and g :: "'b"   and n :: "nat"   and ss :: "'a list"   and ts :: "'a list"   and P :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool" assumes "(f, m) \<in> F"   and "(g, n) \<in> F"   and "length ss = m"   and "length ts = n"   and "\<forall>s\<in>set ss. s \<in> trees"   and "\<forall>t\<in>set ts. t \<in> trees"   and "P (f, m) (g, n)"   and "list_emb (emb P) ss ts" shows "emb P (mk f ss) (mk g ts)"
fin_dim_span: fixes A :: "'b vec set"   and n :: "nat" assumes "finite A"   and "A \<subseteq> carrier (module_vec TYPE('b) n)" shows "vectorspace.fin_dim class_ring (module_vec TYPE('b) n\<lparr>carrier := LinearCombinations.module.span class_ring (module_vec TYPE('b) n) A\<rparr>)"
good_imp_qGood_pick: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "good X" shows "qGood (pick X)"
predicate1I: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "P \<le> Q"
simps_27: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat" shows "(case Rrep x21 x22 x23 x24 x25 of Rreq x xa xb xc xd xe xf \<Rightarrow> f1 x xa xb xc xd xe xf | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f2 x21 x22 x23 x24 x25"
conj_idem: fixes a :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<iinter> a = a"
intro: 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 dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and Lq :: "'c"   and enc :: "'c \<Rightarrow> 'b"   and N :: "'c"   and S :: "'c"   and P :: "'c" assumes "Deduct2_with_PseudoOrder var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv Lq"   and "Repr_Neg var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv enc N"   and "Repr_SelfSubst var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc S"   and "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P" shows "Rosser_Form var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi fls prv bprv Lq dsj enc N S P"
cos_coeff_def: shows "cos_coeff = (\<lambda>n. if even n then (- 1) ^ (n div 2) / fact n else 0)"
row_def: fixes i :: "'c"   and A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "row i A = (\<chi>j. A $h i $h j)"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier H'" shows "inv\<^bsub>H'\<^esub> x \<in> carrier H'"
simps_7: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "'a acom"   and vd :: "'a acom" shows "(v ::= va {vb} \<le> vc;; vd) = False"
maxcoeff_ismaxcoeff: fixes t :: "MIR.num" shows "ismaxcoeff t (maxcoeff t)"
findMin_def: fixes q :: "('c, 'd) BinomialHeap" shows "findMin q \<equiv> BinomialHeapStruc.findMin (Rep_BinomialHeap q)"
set_to_map_empty: shows "set_to_map {} = Map.empty"
is_valid_vars_2: shows "list_all (\<lambda>v. abs_range_map v \<noteq> None) abs_ast_variable_section"
inc_dot_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and x :: "item" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "inc_dot 0 x = x"
noteq_le_less: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "m \<noteq> n" shows "m < n"
X: shows "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
induct: fixes x1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and x2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and Pa :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "x1 \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> x2"   and "\<And>lc lc'. (case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) lc' \<Longrightarrow> Pa lc lc'"   and "\<And>lc lc'. lc \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> lc' \<Longrightarrow> Pa lc lc'" shows "Pa x1 x2"
proj_orig_basis_init_syzygy_list: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "proj_orig_basis (length bs) (init_syzygy_list bs) = bs"
inter_aform_plane_ortho_nres_def: fixes p :: "nat"   and Z :: "'b \<times> 'b pdevs"   and n :: "'b"   and g :: "real" shows "inter_aform_plane_ortho_nres p Z n g = RETURN (inter_aform_plane_ortho p Z n g)"
homotopy_eqv_trans: fixes X :: "'a topology"   and Y :: "'b topology"   and U :: "'c topology" assumes "X homotopy_equivalent_space Y"   and "Y homotopy_equivalent_space U" shows "X homotopy_equivalent_space U"
mset_le_irrefl: fixes M :: "'a multiset"   and R :: "bool" assumes "M < M" shows "R"
IMP_def: fixes x2 :: "ty"   and x1 :: "ty" shows "x2 IMP x1 \<equiv> Abs_ty (IMP_Rep (Rep_ty x2) (Rep_ty x1))"
adj_reachable_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<rightarrow>\<^bsub>del_arc u\<^esub> b"   and "b \<rightarrow>\<^sup>*\<^bsub>del_arc u\<^esub> c" shows "a \<rightarrow>\<^sup>*\<^bsub>del_arc u\<^esub> c"
assms_2: shows "\<forall>op'\<in>set ops'. op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)"
simps_2: fixes uu :: "int" shows "(Num_arc uu < MinInfty) = False"
tOp_mHom_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" shows "tOp_mHom R M N f g = (\<lambda>x\<in>carrier M. f x \<plusminus>\<^bsub>N\<^esub> g x)"
card_ge_0_finite: fixes A :: "'c set" assumes "0 < card A" shows "finite A"
iAll_True: fixes I :: "nat set" shows "\<box> t I. True"
equals_supr_Above: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and a :: "'a" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "a \<in> Above r B"   and "\<And>a'. a' \<in> Above r B \<Longrightarrow> (a, a') \<in> r" shows "a = wo_rel.supr r B"
AX0_def: shows "AX0 \<equiv> {x. AX0p x}"
ptensor_vec_dim: fixes v1 :: "'b vec"   and v2 :: "'b vec" shows "dim_vec (ptensor_vec v1 v2) = d0"
restrict_eq: fixes A :: "'c set" shows "restrictA A = filter (\<lambda>(k, v). k \<in> A)"
Q_hit_v0_v1: shows "v0 \<leadsto>Q_hit\<leadsto>\<^bsub>H_x\<^esub> v1"
that: fixes \<delta> :: "real" assumes "0 < \<delta>"   and "\<And>c' d'. \<lbrakk>c' \<in> cbox a b; d' \<in> cbox a b; norm (c' - c) \<le> \<delta>; norm (d' - d) \<le> \<delta>\<rbrakk> \<Longrightarrow> norm (integral (cbox c' d') f - integral (cbox c d) f) < \<epsilon>" shows "thesis"
length_rbtreeify_f: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> length kvs" shows "length (snd (rbtreeify_f n kvs)) = length kvs - n"
fun_upd_triv: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" shows "f(x := f x) = f"
igSwapIGVar_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSwapIGVar MOD \<equiv> \<forall>zs z1 z2 xs x. igSwap MOD zs z1 z2 (igVar MOD xs x) = igVar MOD xs (x @xs[z1 \<and> z2]_zs)"
cx1: shows "sum cx {0..<length Vs} = (1::'a)"
subgroup_conjugation_is_inj: fixes g :: "'a"   and H1 :: "'a set"   and H2 :: "'a set" assumes "g \<in> carrier G"   and "H1 \<subseteq> carrier G"   and "H2 \<subseteq> carrier G"   and "g <# H1 #> inv g = g <# H2 #> inv g" shows "H1 = H2"
delta_remove: fixes S :: "'a set"   and a :: "'a"   and b :: "'a \<Rightarrow> 'b"   and c :: "'a \<Rightarrow> 'b" assumes "finite S" shows "(\<Prod>k\<in>S. if k = a then b k else c k) = (if a \<in> S then b a * prod c (S - {a}) else prod c (S - {a}))"
action_obs_change_prefix_eq: fixes n :: "enat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat" assumes "ltake n E = ltake n E'"   and "enat a < n" shows "action_obs E a = action_obs E' a"
s_g: fixes x :: "'a" shows "sets (g x) = sets (count_space UNIV)"
ENC_Not: fixes n :: "nat"   and \<phi> :: "'b formula" shows "ENC n (FOV (FNot \<phi>)) = ENC n (FOV \<phi>)"
lincomb_concat: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and rss :: "'a list list"   and mss :: "'b list list" assumes "VectorSpace smult V"   and "list_all2 (\<lambda>rs ms. length rs = length ms) rss mss" shows "scalar_mult.lincomb smult (concat rss) (concat mss) = sum_list (map2 (scalar_mult.lincomb smult) rss mss)"
star_of_approx_iff: fixes x :: "'b"   and y :: "'b" shows "(star_of x \<approx> star_of y) = (x = y)"
convergentI: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c" assumes "X \<longlonglongrightarrow> L" shows "convergent X"
if_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'d"   and u :: "'d"   and y :: "'d"   and v :: "'d" assumes "b = c"   and "c \<Longrightarrow> x = u"   and "\<not> c \<Longrightarrow> y = v" shows "(if b then x else y) = (if c then u else v)"
base_case_satisfy_equation_alt: fixes qs :: "real poly list"   and p :: "real poly" assumes "length qs = 1"   and "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "satisfy_equation p qs [[], [0]] [[1], [- 1]]"
succ_inject_iff: fixes i :: "V"   and j :: "V" shows "(ZFC_in_HOL.succ i = ZFC_in_HOL.succ j) = (i = j)"
absorb_iff2: fixes b :: "nat"   and a :: "nat" shows "(b dvd a) = (gcd a b = b)"
remove1_append: fixes x :: "'g"   and xs :: "'g list"   and ys :: "'g list" shows "remove1 x (xs @ ys) = (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
simps_7: fixes t :: "'a local.term" shows "\<^bold>\<lfloor>\<^bold>\<r>\<^bold>[t\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
fetch_instr_result_2: fixes s' :: "'b sparc_state" assumes "\<exists>v. fetch_instruction s' = Inr v" shows "\<nexists>e. fetch_instruction s' = Inl e"
update_convs_2: fixes values' :: "(vn \<Rightarrow> val option) \<Rightarrow> vn \<Rightarrow> val option"   and tag :: "obj_tag"   and "values" :: "vn \<Rightarrow> val option"   and more :: "'a" shows "values_update values' \<lparr>tag = tag, values = values, \<dots> = more\<rparr> = \<lparr>tag = tag, values = values' values, \<dots> = more\<rparr>"
spectral_radius_1_jnf_max_def: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list" shows "spectral_radius_1_jnf_max A n m lam n_as \<equiv> (A \<in> carrier_mat n n \<and> nonneg_mat A \<and> jordan_nf (of_real_hom.mat_hom A) n_as) \<and> ((m, lam) \<in> set n_as \<and> cmod lam = 1) \<and> (\<forall>x. poly (char_poly A) x = 0 \<longrightarrow> x \<le> 1) \<and> (\<forall>k la. (k, la) \<in> set n_as \<longrightarrow> cmod la \<le> 1 \<and> (cmod la = 1 \<longrightarrow> k \<le> m))"
a_objects_exist_2: fixes v :: "i" shows "[\<^bold>\<box>\<^bold>\<not>(\<^bold>\<forall>x. \<lparr>O!,x\<^sup>P\<rparr>) in v]"
refines_def: fixes e :: "'b \<times> ('c \<times> 'b) list"   and s :: "'d"   and a :: "'c"   and t :: "'d"   and A :: "('b, 'c) ioa"   and f :: "'d \<Rightarrow> 'b" shows "refines e s a t A f \<equiv> fst e = f s \<and> last_state e = f t \<and> is_exec_frag_of A e \<and> (let tr = trace (ioa.asig A) e in if a \<in> externals (ioa.asig A) then tr = [a] else tr = [])"
simps_2: fixes A :: "'a mat"   and I :: "nat" shows "swap_row_to_front A (Suc I) = swap_row_to_front (swaprows I (Suc I) A) I"
inject_7: fixes x7 :: "('a, 'b, 'c) expr"   and y7 :: "('a, 'b, 'c) expr" shows "(Rfork x7 = Rfork y7) = (x7 = y7)"
intros_2: fixes ys :: "(real \<times> real) list"   and xa :: "real \<times> real"   and x :: "real \<times> real" assumes "\<forall>y\<in>set ys. ccw' xa x y"   and "ccw'.sortedP xa ys" shows "ccw'.sortedP xa (x # ys)"
vertical_composite_axioms: shows "vertical_composite A B F G G \<tau> G"
merge_guards_simps_6: fixes b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "merge_guards (While b c) = While b (merge_guards c)"
local_order_carrier_closed: fixes e1 :: "('msgid \<times> 'op) event"   and i :: "nat"   and e2 :: "('msgid \<times> 'op) event" assumes "e1 \<sqsubset>\<^sup>i e2" shows "{e1, e2} \<subseteq> set (history i)"
transfer: fixes b :: "'g" shows "rel_fun (pcr_tllist (=) (=)) (rel_fun (pcr_tllist (=) (=)) (=)) (\<lambda>(xs1, b1) (xs2, b2). if lfinite xs1 then b1 = b \<and> lprefix xs1 xs2 \<or> xs1 = xs2 \<and> flat_ord b b1 b2 else xs1 = xs2) (tllist_ord b)"
that: fixes N :: "nat" assumes "\<forall>n\<ge>N. f n \<in> S" shows "thesis"
0: fixes x :: "real" assumes "x \<in> {0..1}" shows "(\<Sum>b\<in>basis. F c \<bullet> b * (vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b)) = 0"
auxlist'_def: shows "auxlist' \<equiv> case auxlist of [] \<Rightarrow> [(nt, X)] | (t, y) # ts \<Rightarrow> if t = nt then (t, y \<union> X) # ts else (nt, X) # auxlist"
assms: shows "I \<subseteq> {..<brn (cl ! n)}"
typeBinOp2: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and e1 :: "expr"   and e2 :: "expr"   and lev :: "secLevel"   and bop :: "bop" assumes "\<Gamma> \<turnstile> e1 : High"   and "\<Gamma> \<turnstile> e2 : lev" shows "\<Gamma> \<turnstile> e1 \<guillemotleft>bop\<guillemotright> e2 : High"
path_valid_node_2: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n as n'" shows "CFGExit_wf.valid_node n'"
matrix_vector_mult_diff_rdistrib: fixes A :: "(('a, 'c) vec, 'b) vec"   and B :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "(A - B) *v x = A *v x - B *v x"
proj_def: fixes r :: "('b \<times> 'a) set"   and x :: "'b" shows "Equiv_Relations.proj r x = r `` {x}"
shared_def: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat" shows "shared t var = (\<forall>st\<^sub>1 st\<^sub>2. st\<^sub>1 \<le> t \<and> st\<^sub>2 \<le> t \<longrightarrow> isomorphic_dags_eq st\<^sub>1 st\<^sub>2 var)"
component_of_term_splus: fixes t :: "'a"   and v :: "'t" shows "component_of_term (t \<oplus> v) = component_of_term v"
Asm: shows "\<forall>n. Pr n \<longrightarrow> n < n2 \<longrightarrow> Pr (Suc n)"
shiftr_eq_0: fixes n :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n" shows "w >> n = 0"
abs_minus_commute: 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 "abs (minus a b) = abs (minus b a)"
minus_add_distrib: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "- (x + y) = - x + - y"
has_codomain_iff_arr: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "(codomains f \<noteq> {}) = arr f"
CondThrow: fixes e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>if (throw e) e\<^sub>1 else e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
bound_atoms_def: fixes s :: "('b, 'a) state" shows "\<B>\<^sub>A s = (\<lambda>(v, x). Geq v x) ` set_of_map (\<B>\<^sub>l s) \<union> (\<lambda>(v, x). Leq v x) ` set_of_map (\<B>\<^sub>u s)"
to': shows "h \<turnstile> to_tree_order ptr' \<rightarrow>\<^sub>r to'"
residue_primroot_is_generator: fixes m :: "nat"   and g :: "nat" assumes "1 < m"   and "residue_primroot m g" shows "bij_betw (\<lambda>i. g ^ i mod m) {..<totient m} (totatives m)"
extremum_uniqueI: fixes a :: "'b" assumes "1 \<le> a" shows "a = 1"
less_enat_iff: fixes a :: "enat"   and i :: "nat" shows "(a < enat i) = (\<exists>j. a = enat j \<and> j < i)"
prv_imp_monoL: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<chi> \<psi>)" shows "prv (imp (imp \<phi> \<chi>) (imp \<phi> \<psi>))"
locally_trans_def: fixes R :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" shows "locally_trans R A B C \<equiv> \<forall>t u v. t \<in># A \<longrightarrow> u \<in># B \<longrightarrow> v \<in># C \<longrightarrow> (t, u) \<in> R \<longrightarrow> (u, v) \<in> R \<longrightarrow> (t, v) \<in> R"
lift_set_Un_distrib: fixes i :: "nat"   and A :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and B :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" shows "lift_set i (A \<union> B) = lift_set i A \<union> lift_set i B"
ltln_of_dnf_semantics: fixes \<A> :: "'a ltln fset set"   and w :: "nat \<Rightarrow> 'a set" assumes "finite \<A>" shows "w \<Turnstile>\<^sub>n ltln_of_dnf \<A> = (\<exists>\<Phi>\<in>\<A>. \<forall>\<phi>. \<phi> |\<in>| \<Phi> \<longrightarrow> w \<Turnstile>\<^sub>n \<phi>)"
integral_Un: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'a set" assumes "f integrable_on S"   and "f integrable_on T"   and "negligible (S \<inter> T)" shows "integral (S \<union> T) f = integral S f + integral T f"
permutation_self_in_orbit: fixes f :: "'e \<Rightarrow> 'e"   and s :: "'e" assumes "permutation f" shows "s \<in> orbit f s"
FINITE_states: fixes X :: "'a set" assumes "finite X" shows "finite {s. fmdom' s = X}"
subset_eq_atLeast0_atMost_finite: fixes N :: "nat set"   and n :: "nat" assumes "N \<subseteq> {0..n}" shows "finite N"
find_indices_set: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "set (find_indices P l) = {i. i < length l \<and> P (l ! i)}"
rbl_plus_def: fixes cin :: "bool"   and xs :: "bool list"   and ys :: "bool list" shows "rbl_plus cin xs ys = foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (\<lambda>_. []) cin"
simps_2: fixes f :: "'a"   and ss :: "('a, 'c) Term.term list"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "Fun f ss \<cdot> \<sigma> = Fun f (ss \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<sigma>)"
rel_with_target_impl_reflC_transC_TRel_is_weak_reduction_contrasimulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_contrasimulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>*" shows "weak_reduction_contrasimulation (TRel\<^sup>*) Target"
discr_ZObis: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.discr tval aval indis c"   and "PL_Indis.discr tval aval indis d" shows "(c, d) \<in> PL_Indis.ZObis tval aval indis"