hcju/isabelleps huggingface dataset:
mapM_None: fixes f :: "'f \<Rightarrow> 'e option"   and xs :: "'f list" shows "(mapM f xs = None) = (\<exists>x\<in>set xs. f x = None)"
prems_1: shows "u \<turnstile>\<^bsub>v,n\<^esub> m"
atUV_single: fixes U :: "'f set"   and V :: "'f set"   and s :: "'f list" shows "\<not> atUV U V {#s#}"
eq_span_insert_eq: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "x - y \<in> pmdl S" shows "pmdl (insert x S) = pmdl (insert y S)"
list_mwb_lens: fixes x :: "nat" shows "mwb_lens (list_lens x)"
co_path_simps_4: fixes e1 :: "'b \<times> 'b"   and e :: "'b \<times> 'b"   and w :: "'b"   and e2 :: "'b \<times> 'b"   and es :: "('b \<times> 'b) list" assumes "e1 \<noteq> (fst e, w) \<or> e2 \<noteq> (w, snd e)"   and "e1 \<noteq> (snd e, w) \<or> e2 \<noteq> (w, fst e)" shows "co_path e w (e1 # e2 # es) = e1 # co_path e w (e2 # es)"
not_less_inf: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.top_weight less_eq less top plus zero" shows "(\<not> less a top) = (a = top)"
le_st_def: fixes F :: "'b Abs_State.st"   and G :: "'b Abs_State.st" shows "(F \<sqsubseteq> G) = (\<forall>x\<in>set (Abs_State.dom G). lookup F x \<sqsubseteq> fun G x)"
lossless_assert_spmf: fixes b :: "bool" shows "lossless_spmf (assert_spmf b) = b"
is_standard_operator_rem_implicit_pres_ops': fixes op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "op \<in> set (ast_problem.ast\<delta> (rem_implicit_pres_ops prob))"   and "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> is_standard_operator op" shows "is_standard_operator op"
asIMOD_igSwapInp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and inp :: "'c \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b) option" assumes "sWlsVal SEM val" shows "lift (\<lambda>X. X val) (igSwapInp (asIMOD SEM) zs z1 z2 inp) = lift (\<lambda>X. X (val ^[z1 \<and> z2]_zs)) inp"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) ((\<mu> \<star> \<nu>) \<star> \<pi>)"
assms: shows "accept\<^sub>G\<^sub>R_LTS R w"
less_2: fixes k :: "nat" assumes "sdrop k w = sconst (w !! k)" shows "thesis"
b_rel_subtyping_merge_eqA_tL: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and P' :: "'a \<Rightarrow> bool" shows "hn_ctxt A x y \<or>\<^sub>A hn_ctxt (b_assn A P') x y \<Longrightarrow>\<^sub>t hn_ctxt A x y"
jpTraces_def: shows "jpTraces \<equiv> \<lambda>jp. {x. jpTracesp jp x}"
hyps_4: assumes "\<And>xs. rtrancl_path r y_ xs z_ \<Longrightarrow> thesis" shows "thesis"
wte: shows "\<Gamma> \<turnstile> e : \<tau>"
shift1_def: fixes p :: "poly" shows "shift1 p = CN 0\<^sub>p 0 p"
SdlHPPRV: shows "SdlHPPB d PP RV"
eval\<Pi>\<^sub>2_inverse: fixes x :: "\<Pi>\<^sub>2" shows "make\<Pi>\<^sub>2 (eval\<Pi>\<^sub>2 x) = x"
p_residue_alt_def: fixes m :: "nat"   and n :: "int" shows "p_residue m n = n mod p ^ m"
bisims1_ThrowD: fixes P :: "'b prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and vs :: "'addr val list"   and a :: "'addr"   and es' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,es,h \<turnstile> (map Val vs @ Throw a # es', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)" shows "pc < length (compEs2 es) \<and> (xcp = \<lfloor>a\<rfloor> \<or> xcp = None) \<and> xs = loc"
dX: fixes X :: "real \<Rightarrow> 'a" assumes " D X = (\<lambda>\<tau>. f \<tau> (X \<tau>)) on U (X t\<^sub>0)" shows " D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (\<lambda>\<tau>. \<mu>' (X \<tau>) - \<nu>' (X \<tau>)) on U (X t\<^sub>0)"
circ_transitive_equal: fixes x :: "'a" shows "idempotent (x\<^sup>\<circ>)"
ratrel_equivp: shows "part_equivp (\<approx>)"
Sup_sup: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<forall>X. f (\<Squnion>X) = \<Squnion>{f x |x. x \<in> X}" shows "f (x \<squnion> y) = f x \<squnion> f y"
compl_struct_f4_red: shows "punit.compl_struct punit.f4_red"
len_signed: fixes x :: "'a signed itself" shows "len_of x = LENGTH('a)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (\<pi> \<star> \<rho>)"
ideal_mult_right_in2: fixes a :: "'a"   and A :: "'a set"   and b :: "'a" assumes "a \<in> More_Modules.ideal A" shows "b * a \<in> More_Modules.ideal A"
S_def: shows "S = norm (a /\<^sub>R (1 - b))"
compose_Tangle_def: fixes x :: "Tangle_Diagram"   and y :: "Tangle_Diagram" shows "x \<circ> y = Abs_Tangle_Diagram (Rep_Tangle_Diagram x \<circ> Rep_Tangle_Diagram y)"
fixp_induct: fixes b :: "'b"   and P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" assumes "ccpo.admissible (flatf_lub b) (flatf_ord b) P"   and "flatf_mono b f"   and "P (flatf_lub b {})"   and "\<And>x. P x \<Longrightarrow> P (f x)" shows "P (flatf_fp b f)"
relcomp_on_triple: fixes R :: "('f \<times> 'h) set"   and S :: "('h \<times> 'g) set" shows "on_triple (R O S) = on_triple R O on_triple S"
associated_sym: fixes a :: "'a list"   and b :: "'a list" assumes "a \<sim>\<^bsub>poly_ring R\<^esub> b" shows "b \<sim>\<^bsub>poly_ring R\<^esub> a"
swap_inj_on: fixes A :: "('c \<times> 'd) set" shows "inj_on (\<lambda>(i, j). (j, i)) A"
tensor0_prod_left: fixes ds :: "nat list"   and A :: "'a tensor" shows "tensor0 ds \<otimes> A = tensor0 (ds @ dims A)"
select_convs_1: fixes nothing :: "unit"   and more :: "'m" shows "RObject.nothing \<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = nothing"
surjective: fixes r :: "('c, 'd, 'e) prism_scheme" shows "r = \<lparr>prism_match = match\<^bsub>r\<^esub>, prism_build = build\<^bsub>r\<^esub>, \<dots> = prism.more r\<rparr>"
select_convs_3: fixes verts :: "'d set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'd"   and head :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "tail \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = tail"
f_closed: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "semigroup_ow U f"   and "a \<in> U"   and "b \<in> U" shows "f a b \<in> U"
tendsto_inverse_ereal: fixes u :: "'b \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'b filter" assumes "(u \<longlongrightarrow> l) F"   and "l \<noteq> 0" shows "((\<lambda>x. 1 / u x) \<longlongrightarrow> 1 / l) F"
intro: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x. f (x + (1::'a)) = f x" shows "periodic_fun_simple' f"
indaut_chmap: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp X (OpposedThinChamberComplexFoldings.induced_automorphism X f g ` C)"
hurwitz_zeta_cnj: assumes "0 < a" shows "hurwitz_zeta a (cnj (complex_of_real x)) = cnj (hurwitz_zeta a (complex_of_real x))"
finite_measure_axioms: fixes n :: "nat" shows "finite_measure (M n)"
rotation_preserve_distance_formula: fixes u :: "complex_homo"   and v :: "complex_homo"   and \<phi> :: "real" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance_formula (to_complex (moebius_pt (moebius_rotation \<phi>) u)) (to_complex (moebius_pt (moebius_rotation \<phi>) v)) = poincare_distance_formula (to_complex u) (to_complex v)"
assms: fixes n :: "nat" shows "degree (p $ n) \<le> d"
Cons_4: fixes g :: "'t \<Rightarrow>\<^sub>0 'b"   and s :: "'a" assumes "g \<in> set (g__ # gs)"   and "p = monom_mult (1::'b) s g" shows "thesis"
SA_all: fixes \<Gamma> :: "binding list"   and T\<^sub>1 :: "POPLmark.type"   and S\<^sub>1 :: "POPLmark.type"   and S\<^sub>2 :: "POPLmark.type"   and T\<^sub>2 :: "POPLmark.type" assumes "\<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1"   and "TVarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2" shows "\<Gamma> \<turnstile> (\<forall><:S\<^sub>1. S\<^sub>2) <: (\<forall><:T\<^sub>1. T\<^sub>2)"
zero: shows "mod_ring_rel (arith_ops_record.zero (finite_field_ops_int p)) 0"
narrow_RefT2: fixes G :: "prog"   and S :: "ty"   and R :: "ref_ty" assumes "G\<turnstile>S\<succ>RefT R" shows "\<exists>t. S = RefT t"
subset_divisors_dvd: fixes a :: "'a"   and b :: "'a" shows "({c. c dvd a} \<subseteq> {c. c dvd b}) = (a dvd b)"
tl: shows "{y EQ y', z EQ z'} \<turnstile> P x' y z IFF P x' y' z'"
less_eq_complex_def: fixes x :: "complex"   and y :: "complex" shows "x \<le> y \<equiv> Im x = Im y \<and> Re x \<le> Re y"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and role :: "role" shows "P (case role of ClientRole \<Rightarrow> f1 | ServerRole \<Rightarrow> f2) = ((role = ClientRole \<longrightarrow> P f1) \<and> (role = ServerRole \<longrightarrow> P f2))"
IH: fixes p :: "int" shows "if p = int n then reduced_homology_group (int n) (nsphere n) \<cong> integer_group else trivial_group (reduced_homology_group p (nsphere n))"
simps_8: fixes f :: "'c \<Rightarrow> 'b" shows "map f \<epsilon> = \<epsilon>"
in_lesI_1: fixes s :: "bdd"   and s' :: "bdd"   and ni1 :: "nat"   and n1 :: "nat ifex"   and ni2 :: "nat"   and n2 :: "nat ifex" assumes "mi.les s s'"   and "(ni1, n1) \<in> Rmi s"   and "(ni2, n2) \<in> Rmi s" shows "(ni1, n1) \<in> Rmi s'"
alpha_Tree_supp_rel: fixes t1 :: "('a, 'b, 'c) Formula.Tree"   and t2 :: "('a, 'b, 'c) Formula.Tree" assumes "t1 =\<^sub>\<alpha> t2" shows "supp_rel (=\<^sub>\<alpha>) t1 = supp_rel (=\<^sub>\<alpha>) t2"
affine_hull_eq: fixes s :: "'b set" shows "(affine hull s = s) = affine s"
Efin_imp_Vfin: assumes "finite cf.E" shows "finite cf.V"
degree_mult_right_le: fixes q :: "'a poly"   and p :: "'a poly" assumes "q \<noteq> 0" shows "degree p \<le> degree (p * q)"
strict_trans1: fixes b :: "'d"   and a :: "'d"   and c :: "'d" assumes "b \<le> a"   and "c < b" shows "c < a"
fpxs_nth_minus: fixes f :: "'b fpxs"   and g :: "'b fpxs"   and r :: "rat" shows "fpxs_nth (f - g) r = fpxs_nth f r - fpxs_nth g r"
oexp_Partial_order: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "wo_rel2 r s" shows "Partial_order (r ^o s)"
unique: fixes x :: "nat \<times> nat"   and y :: "nat"   and z :: "nat" assumes "P x y"   and "P x z" shows "y = z"
axioms_3: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e" assumes "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
reds_smaller_L: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp"   and L' :: "Vars.var list" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z"   and "set L' \<subseteq> set L" shows "\<Gamma> : e \<Down>\<^bsub>L'\<^esub> \<Delta> : z"
Arg2pi_def: fixes z :: "complex" shows "Arg2pi z \<equiv> if z = 0 then 0 else THE t. 0 \<le> t \<and> t < 2 * pi \<and> is_Arg z t"
fun_upd_other: fixes z :: "'d"   and x :: "'d"   and f :: "'d \<Rightarrow> 'e"   and y :: "'e" assumes "z \<noteq> x" shows "(f(x := y)) z = f z"
comp_id: fixes f :: "'e \<Rightarrow> 'f" shows "f \<circ> id = f"
f1_2: shows "\<forall>i j. \<phi> (the (f\<^sub>1 i j)) = local.adverse 1 i j"
selectlenhelp2: fixes i :: "nat" shows "\<forall>l j. \<exists>k. select l i j = select k 0 (j - i)"
lock_ok_las'_collect_locks'_may_lock: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions"   and l :: "'c" assumes "lock_ok_las' ls t las"   and "\<forall>l\<in>collect_locks' las. may_lock (ls $ l) t"   and "l \<in> collect_locks las" shows "may_lock (ls $ l) t"
dims0_def: shows "st3.dims0 = nths dims st3.vars0"
cases: fixes a :: "'a"   and r :: "('b \<times> 'a) set" assumes "a \<in> Range r" obtains a :: "'b" and b :: "'a" where "aa__ = b" and "(a, b) \<in> r"
A22_DAcc_level1: shows "DAcc level1 sA22 = {sA31}"
redT_updT'_None: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and ta :: "('h, 'i, 'j) new_thread_action"   and t :: "'h" assumes "redT_updT' ts ta t = None" shows "ts t = None"
x_y_props_1: shows "(x, y) \<in> induced_phi_graph g P"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H G" shows "H \<subseteq> carrier G"
seq_distrib_right_axioms: shows "seq_distrib_right (;) nil"
ker_def: shows "kerT = {v \<in> carrier V. T v = \<zero>\<^bsub>W\<^esub>}"
infpath_length: fixes f :: "nat \<Rightarrow> 'a list"   and n :: "nat" assumes "wf_infpath f" shows "length (f n) = n"
comp_def: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "f \<circ> g = (\<lambda>x. f (g x))"
subseteq_guards_Seq: fixes c :: "('d, 'e, 'f) com"   and c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g Seq c1 c2" shows "\<exists>c1' c2'. c = Seq c1' c2' \<and> (c1' \<subseteq>\<^sub>g c1) \<and> (c2' \<subseteq>\<^sub>g c2)"
finite_vars_mset: fixes A :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) multiset" shows "finite (vars_mset A)"
h3: shows "M_mat (take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) (take_indices subsets (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))))) = take_rows_from_matrix (take_cols_from_matrix (M_mat signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) (rows_to_keep (take_cols_from_matrix (M_mat signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))))"
a: shows "R[x\<turnstile>n>y] = OrL v.M w.N u"
arr_UP: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "arr (UP \<mu>)"
induct: fixes x1 :: "super_binop + boolean_binop + numeric_binop + string_binop + collection_binop"   and x2 :: "call_kind"   and x3 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type"   and x5 :: "'a OCL_Types.type"   and P :: "super_binop + boolean_binop + numeric_binop + string_binop + collection_binop \<Rightarrow> call_kind \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "binop_type x1 x2 x3 x4 x5"   and "\<And>op \<tau> \<sigma> \<rho>. super_binop_type op \<tau> \<sigma> \<rho> \<Longrightarrow> P (Inl op) DotCall \<tau> \<sigma> \<rho>"   and "\<And>op \<tau> \<sigma> \<rho>. boolean_binop_type op \<tau> \<sigma> \<rho> \<Longrightarrow> P (Inr (Inl op)) DotCall \<tau> \<sigma> \<rho>"   and "\<And>op \<tau> \<sigma> \<rho>. numeric_binop_type op \<tau> \<sigma> \<rho> \<Longrightarrow> P (Inr (Inr (Inl op))) DotCall \<tau> \<sigma> \<rho>"   and "\<And>op \<tau> \<sigma> \<rho>. string_binop_type op \<tau> \<sigma> \<rho> \<Longrightarrow> P (Inr (Inr (Inr (Inl op)))) DotCall \<tau> \<sigma> \<rho>"   and "\<And>op \<tau> \<sigma> \<rho>. collection_binop_type op \<tau> \<sigma> \<rho> \<Longrightarrow> P (Inr (Inr (Inr (Inr op)))) ArrowCall \<tau> \<sigma> \<rho>" shows "P x1 x2 x3 x4 x5"
secure_implies_refusal_consistency: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list"   and X :: "'a set" assumes "secure P I D"   and "(xs, ys) \<in> rel_ipurge_aux P I D (D ` X)"   and "X \<in> refusals P xs" shows "X \<in> refusals P ys"
add_to_nat_def: fixes x :: "'b"   and y :: "'b" shows "x + y = mod_type_class.from_nat (mod_type_class.to_nat x + mod_type_class.to_nat y)"
e_fac: shows "\<forall>w\<in>cball z e. f w = g w * (w - z) ^ nat zn \<and> g w \<noteq> 0"
trclp_trcl_eq: fixes t :: "('a \<times> 'b \<times> 'a) set" shows "trclp (\<lambda>x xa xb. (x, xa, xb) \<in> t) = (\<lambda>x xa xb. (x, xa, xb) \<in> trcl t)"
linear_inj_imp_surj: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'c) vec" assumes "linear (*s) (*s) f"   and "inj f" shows "surj f"
method_exit_def: fixes n :: "'b LDCFG_node" shows "local.CFGExit.method_exit n \<equiv> n = NewExit \<or> (\<exists>a Q p f. n = src a \<and> lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a \<and> knd a = Q\<hookleftarrow>\<^bsub>p\<^esub>f)"
r_eq: fixes x :: "nat"   and y :: "nat" shows "eval r_eq [x, y] \<down>= (if x = y then 0 else 1)"
hyps: shows "coeff ([:a, 1::'a:] ^ n) n = (1::'a)"
norm_pre_pure_iff: fixes P :: "assn"   and b :: "bool"   and c :: "'c Heap"   and Q :: "'c \<Rightarrow> assn" shows "<P * \<up> b> c <Q> = (b \<longrightarrow> <P> c <Q>)"
nlpaux: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "x \<notin> dom (Cp b)" shows "Cp (a \<oplus> b) x = Cp a x"
insert_sort_insert_by_set: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and e :: "'a"   and l :: "'a list" shows "set (insert_sort_insert_by cmp e l) = insert e (set l)"
or_att_fst_sup0: fixes x2 :: "'a attree list"   and x :: "'a set \<times> 'a set" assumes "x2 \<noteq> []"   and "\<turnstile>x2 \<oplus>\<^sub>\<or>\<^bsup>x\<^esup>" shows "fst x \<subseteq> (\<Union>y\<in>set x2. fst (attack y))"
convex_same_rel_interior_closure: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "convex T" shows "(rel_interior S = rel_interior T) = (closure S = closure T)"
erfc_of_real: fixes x :: "real" shows "erfc (of_real x) = of_real (erfc x)"
rtNode_2: fixes p :: "ref"   and no :: "ref"   and not :: "dag" assumes "Dag p low high rt"   and "Dag no low high not"   and "no \<in> set_of rt" shows "not \<le> rt"
ptr_add_def: fixes ptr :: "'b word"   and n :: "nat" shows "ptr_add ptr n \<equiv> ptr + word_of_nat n"
hs_new_impl: shows "imp_set_empty is_hashset hs_new"
construct_outside: fixes B :: "'b set"   and v :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "vs1.independent B"   and "v \<in> vs1.span (vs1.extend_basis B - B)" shows "construct B f v = (0::'c)"
DERIV_local_max: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real"   and d :: "real" assumes "(f has_real_derivative l) (at x)"   and "0 < d"   and "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x" shows "l = 0"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<^bold>\<lfloor>t \<^bold>\<otimes> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>"
Pp: shows "extreme_bound A (\<sqsubseteq>) P p"
that: fixes d :: "real"   and B :: "real" assumes "0 < d"   and "0 < B"   and "\<And>x y. \<lbrakk>x \<in> {a--b}; dist x y \<le> d\<rbrakk> \<Longrightarrow> B \<le> f y \<bullet> rot (a - b)" shows "thesis"
toList_difference: fixes xs :: "'c odlist"   and ys :: "'c odlist" shows "toList (ODList.difference xs ys) = List_local.difference (toList xs) (toList ys)"
permutes_inv: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "inv p permutes S"
length_append: fixes xs :: "'m list"   and ys :: "'m list" shows "length (xs @ ys) = length xs + length ys"
urel_integer_mod: fixes pp :: "integer"   and x :: "integer"   and x' :: "int"   and y :: "integer"   and y' :: "int" assumes "p = int_of_integer pp"   and "urel_integer x x'"   and "urel_integer y y'" shows "urel_integer (if y = 0 then x else 0) (if y' = 0 then x' else 0)"
bij_betw_inv: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "\<exists>g. bij_betw g B A"
dropWhile_sing_pow: fixes a :: "'b"   and w :: "'b list" shows "(dropWhile (\<lambda>x. x = a) w = \<epsilon>) = (w = [a] \<^sup>@ \<^bold>|w\<^bold>|)"
regular2: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "regular (mk_regular2 P F)"
gkernTr3: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "x \<in> gker\<^bsub>F,G\<^esub> f" shows "\<rho>\<^bsub>F\<^esub> x \<in> gker\<^bsub>F,G\<^esub> f"
subst_compose: fixes \<sigma> :: "'c \<Rightarrow> ('a, 'd) Term.term"   and \<tau> :: "'d \<Rightarrow> ('a, 'b) Term.term"   and x :: "'c" shows "(\<sigma> \<circ>\<^sub>s \<tau>) x = \<sigma> x \<cdot> \<tau>"
chamber_distance_le: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "folding.gallery (C # Cs @ [D])" shows "order.greater_eq (Suc (length Cs)) (chamber_distance C D)"
is_map: shows "is_left_adjoint TTfgh_THfgh.chine"
lossless: fixes s :: "'s" assumes "guard s"   and "I s" shows "lossless_spmf (body s)"
exec_as_proj_valid_2: fixes a :: "('b, 'c) fmap \<times> ('b, 'd) fmap"   and PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set" assumes "a \<in> PROB" shows "action_dom (fst a) (snd a) \<subseteq> prob_dom PROB"
rec: fixes f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and x :: "'e \<Rightarrow> 'f" shows "rec_stateT f (StateT x) = f x"
prog_not_eq_in_ctran: fixes P :: "'b com option"   and s :: "'b"   and t :: "'b" shows "((P, s), P, t) \<notin> ctran"
stream_space_Stream: fixes x :: "'b"   and \<omega> :: "'b stream"   and M :: "'b measure" shows "(x ## \<omega> \<in> space (stream_space M)) = (x \<in> space M \<and> \<omega> \<in> space (stream_space M))"
3: shows "row_length A1 * row_length B1 = row_length (A1 \<circ> A2 \<otimes> B1 \<circ> B2)"
cond_monotonic: fixes P :: "'d \<Rightarrow> ('e, 'f) uexpr"   and Q :: "'d \<Rightarrow> ('e, 'f) uexpr"   and b :: "'f upred" assumes "mono P"   and "mono Q" shows "mono (\<lambda>X. P X \<triangleleft> b \<triangleright> Q X)"
r_witness: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and x :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A"   and "x \<in> perron_frobenius.X" shows "\<exists>j. 0 < x $h j \<and> perron_frobenius.r A x = (A *v x) $h j / x $h j"