hcju/isabelleps huggingface dataset:
upto_raw_def: fixes n :: "int"   and m :: "int" shows "upto_raw n m = (if m \<le> n then Yield m (m + 1) else Done)"
QD_invar_def: fixes u :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and Q :: "nat list" shows "QD_invar u f Q \<equiv> u \<in> V - {s, t} \<and> distinct Q \<and> set Q = {v \<in> V - {s, t, u}. excess f v \<noteq> (0::'a)}"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "\<langle>'b\<rangle>\<^sub>\<bottom>" shows "P (case option of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = ((option = \<bottom> \<longrightarrow> P f1) \<and> (\<forall>x2. option = \<lfloor>x2\<rfloor> \<longrightarrow> P (f2 x2)))"
rel_simps_56: fixes n :: "num" shows "- numeral n < (0::'a)"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G local.map"
pathfinish_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "pathfinish (subpath u v g) = g v"
times_Box_def: fixes x :: "'b Box"   and y :: "'b Box" shows "x * y \<equiv> bliftM2 (\<Lambda> a b. a * b)\<cdot>x\<cdot>y"
weak_get_via_put: fixes x :: "'d \<Longrightarrow> 'e"   and s :: "'e" assumes "mwb_lens x"   and "s \<in> \<S>\<^bsub>x\<^esub>" shows "get\<^bsub>x\<^esub> s = (THE v. put\<^bsub>x\<^esub> s v = s)"
inj_imp_image_ring_iso: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme" assumes "inj_on f (carrier R)" shows "f \<in> ring_iso R (image_ring f R)"
map_iterator_abs_genord_I2: fixes iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<times> 'g \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "'f \<Rightarrow> 'g option"   and R' :: "'f \<times> 'g \<Rightarrow> 'f \<times> 'g \<Rightarrow> bool"   and invar :: "'g \<Rightarrow> bool"   and R :: "'f \<times> 'h \<Rightarrow> 'f \<times> 'h \<Rightarrow> bool"   and \<alpha> :: "'g \<Rightarrow> 'h"   and m' :: "'f \<Rightarrow> 'h option" assumes "map_iterator_genord iti m R'"   and "\<And>k v. m k = Some v \<Longrightarrow> invar v"   and "\<And>k v k' v'. \<lbrakk>invar v; invar v'; R' (k, v) (k', v')\<rbrakk> \<Longrightarrow> R (k, \<alpha> v) (k', \<alpha> v')"   and "m' = map_option \<alpha> \<circ> m" shows "map_iterator_abs_genord \<alpha> invar iti m' R"
op_iam_fmap_empty_def: shows "op_iam_fmap_empty = fmempty"
set_in_cone: fixes Vs :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> carrier_vec n" shows "Vs \<subseteq> gram_schmidt.cone n Vs"
face_cycle_sets_def: fixes G :: "('d, 'c) pre_digraph"   and M :: "'c pre_map" shows "pre_digraph_map.face_cycle_sets G M = pre_digraph_map.face_cycle_set M ` arcs G"
simps_2: fixes \<eta> :: "trm option" shows "Timeso undeft \<eta> = undeft"
optimal_set_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" shows "optimal_set f S = {x \<in> S. \<forall>y\<in>S. f x y}"
less_le_not_le: fixes x :: "'b list"   and y :: "'b list" shows "(x <p y) = (x \<le>p y \<and> \<not> y \<le>p x)"
Pd_refl: fixes x :: "'a"   and d :: "'b" assumes "x \<in> Field (Pd d)" shows "(x, x) \<in> Pd d"
case_prod_beta': fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
poly_of_pm_eq_zero_iff: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "(poly_of_pm x p = 0) = (keys p \<inter> .[{x}] = {})"
asymD: fixes R :: "('g \<times> 'g) set"   and x :: "'g"   and y :: "'g" assumes "asym R"   and "(x, y) \<in> R" shows "(y, x) \<notin> R"
is_solution_solve_system: fixes x :: "('b, 'c) vec"   and A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "is_solution x A b = is_solution x (fst (solve_system A b)) (snd (solve_system A b))"
segment_bound1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> closed_segment a b" shows "norm (x - a) \<le> norm (b - a)"
inf_top_right: fixes x :: "'b" shows "x \<sqinter> \<top> = x"
sum: shows "sum_list (map weight F') < sum_list (map weight F)"
someinit_def: fixes i :: "nat" shows "opq.someinit i \<equiv> SOME x. x \<in> (fst \<circ> (\<lambda>(p, q). (fst (Fun.id p), snd (Fun.id p), q))) ` init (ptoy i \<langle>\<langle> qmsg)"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (L a)"
ccomp_comparator_prod: shows "ccomp = comparator_prod ccomp ccomp"
mset_le_single_iff: fixes x :: "'a"   and y :: "'a" shows "({#x#} \<le> {#y#}) = (x \<le> y)"
perm_4: fixes pi :: "(name \<times> name) list"   and x3 :: "trm"   and x1 :: "name"   and x2 :: "trm" shows "pi \<bullet> x3 to x1 in x2 = (pi \<bullet> x3) to pi \<bullet> x1 in (pi \<bullet> x2)"
fun_add_eval_closed: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier (function_ring (carrier R) R)"   and "g \<in> carrier (function_ring (carrier R) R)" shows "(f \<oplus>\<^bsub>function_ring (carrier R) R\<^esub> g) a \<in> carrier R"
simps_1: fixes tset1 :: "('a, 'b, 'c, 'd) Tree set['a]"   and tset2 :: "('a, 'b, 'c, 'd) Tree set['a]" shows "(tConj tset1 =\<^sub>\<alpha> tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
simps_1: fixes uu :: "('b \<times> 'b) set"   and uv :: "'a \<Rightarrow> 'b"   and uw :: "'b" shows "c_ipurge uu uv uw [] = []"
arg_unique: fixes z :: "complex"   and x :: "real" assumes "sgn z = cis x"   and "- pi < x"   and "x \<le> pi" shows "arg z = x"
antisym: fixes ko :: "'b key_order"   and x :: "'b"   and y :: "'b" assumes "ko.le ko x y"   and "ko.le ko y x" shows "x = y"
Dag_upd_same_r_lemma: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "p \<noteq> Null" shows "\<not> Dag p l (r(p := p)) t"
distinct_8: fixes x51 :: "'b formula"   and x52 :: "'b formula"   and x1 :: "'b" shows "x51 \<^bold>\<or> x52 \<noteq> Atom x1"
card_of_ordLeq_finite: fixes A :: "'c set"   and B :: "'d set" assumes "|A| \<le>o |B|"   and "finite B" shows "finite A"
Restrict_iff: fixes x :: "'d"   and y :: "'e"   and A :: "'d set"   and r :: "('d \<times> 'e) set" shows "((x, y) \<in> Restrict A r) = ((x, y) \<in> r \<and> x \<in> A)"
trans_imp_quasi_trans: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "quasi_trans r"
invimage_vrange_vsubset: fixes r :: "V"   and B :: "V" assumes "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> B" shows "r -`\<^sub>\<circ> B = \<D>\<^sub>\<circ> r"
stream_pred_snth: fixes P :: "'c \<Rightarrow> bool"   and w :: "'c stream" shows "pred_stream P w = (\<forall>i. P (w !! i))"
sep_conj_impl1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "\<And>h. P h \<Longrightarrow> I h"   and "sep_algebra.sep_conj plus sep_disj P R h" shows "sep_algebra.sep_conj plus sep_disj I R h"
step_8: shows "\<alpha> = \<omega>_sum \<beta>s"
asm_3: shows "length (predecessors g n) \<noteq> 1"
tp_upt_Suc: shows "[tp + 1 - (len + 1)..<tp + 1] = [tp - len..<tp] @ [tp]"
integrable_iff_bounded: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>)"
continuous_closedin_preimage_constant: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'c" assumes "continuous_on S f" shows "closedin (top_of_set S) {x \<in> S. f x = a}"
leg0_in_hom_1: shows "\<guillemotleft>r\<^sub>0 : src r\<^sub>0 \<rightarrow> src r\<guillemotright>"
num_ni_def: fixes ts :: "nat \<Rightarrow> thread" shows "num_ni ts \<equiv> \<Sum>i = 0..<N. ni_weight (ts i)"
const: fixes c :: "'b"   and F :: "'a filter" assumes "c \<noteq> (0::'b)" shows "L F (\<lambda>_. c) = L F (\<lambda>_. 1::'b)"
inverse_zero_imp_zero: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "inverse a = zero" shows "a = zero"
simps_2: fixes uu :: "val" shows "free_vars (CVal uu) = {}"
span_some_basis_supset: fixes V :: "'b set" shows "V \<subseteq> local.span (some_basis V)"
iMODb2_set_iMODb_set_subset: shows "iMODb2_set \<subseteq> iMODb_set"
drop_append: fixes n :: "nat"   and xs :: "'e list"   and ys :: "'e list" shows "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
FORKs_def2: shows "fold (\<lambda>i P. P|||FORK i) [1..<N] (FORK 0) = fold (\<lambda>i P. P|||FORK i) [0..<N] SKIP"
frag_of_eq: fixes x :: "'c"   and y :: "'c" shows "(frag_of x = frag_of y) = (x = y)"
H2N_inv: fixes b :: "nat"   and n :: "nat" shows "H2N (Suc (Suc b)) (N2H (Suc (Suc b)) n) = n"
simps_3: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> (X\<^sub>n \<phi>) w = \<G>\<F> \<phi> w"
Block_def: shows "Block = (\<lambda>a\<in>S. THE A. a \<in> A \<and> A \<in> P)"
treewidth_cards_nonempty: shows "treewidth_cards \<noteq> {}"
hyps: shows "degree (prod_list ps) \<le> sum_list (map degree ps)"
hull_redundant: fixes a :: "'d"   and S :: "'d set \<Rightarrow> bool"   and s :: "'d set" assumes "a \<in> S hull s" shows "S hull insert a s = S hull s"
IH_2: fixes x :: "'a \<times> 'b"   and xa :: "('a \<times> 'b) set"   and xb :: "('a \<times> 'b) set" assumes "\<not> (infinite A \<or> A = {})"   and "x = arg_min_on snd A"   and "xa = {p \<in> A. fst p < fst x}"   and "xb = {p \<in> A. fst x < fst p}" shows "set_tree (treap_of xb) \<subseteq> xb"
assms_5: shows "new \<in> DeclConcepts.imethds G I sig"
3: fixes f :: "('fun, 'var) Term.term \<times> ('fun, 'var) Term.term"   and d :: "('fun, 'var) Term.term \<times> ('fun, 'var) Term.term"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "f \<in> set F'"   and "d \<in> set (map snd D) - set (map snd (dbproj i D))" shows "pair f \<cdot> \<delta> \<noteq> pair d \<cdot> \<delta>"
minus_antichain_def: shows "(-) \<equiv> map_fun set_antichain (map_fun set_antichain antichain) (-)"
compl_coset: fixes xs :: "'b list" shows "- List.coset xs = set xs"
cobounded2: fixes a :: "'b"   and b :: "'b" shows "min a b \<le> b"
nDComm: fixes a :: "'b set set"   and b :: "'b set set" assumes "netsDistinct a b" shows "netsDistinct b a"
assms: shows "((\<sigma>, k), \<sigma>', k') \<in> cop"
oprod_embed: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'"   and "r' \<noteq> {}" shows "embed r (r *o r') (\<lambda>x. (x, minim r' (Field r')))"
create_initial_object_def: shows "create_initial_object = (\<lambda>(h, ads, b) C. if b then let HA = allocate h (Class_type C) in if HA = {} then (h, ads, False) else let (h', a'') = SOME ha. ha \<in> HA in (h', ads @ [a''], True) else (h, ads, False))"
lim_imp_Liminf: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd"   and f0 :: "'d" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> f0) F" shows "Liminf F f = f0"
u_simple_fusion: fixes H :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice"   and F :: "'c::refinement_lattice \<Rightarrow> 'c::refinement_lattice"   and G :: "'b::complete_lattice \<Rightarrow> 'c::refinement_lattice" assumes "mono (H::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)"   and "\<forall>x::'b::complete_lattice. ((F::'c::refinement_lattice \<Rightarrow> 'c::refinement_lattice) \<circ> (G::'b::complete_lattice \<Rightarrow> 'c::refinement_lattice)) x \<sqsubseteq> (G \<circ> (H::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)) x" shows "lfp (F::'c::refinement_lattice \<Rightarrow> 'c::refinement_lattice) \<sqsubseteq> (G::'b::complete_lattice \<Rightarrow> 'c::refinement_lattice) (lfp (H::'b::complete_lattice \<Rightarrow> 'b::complete_lattice))"
order_gt_0: fixes G :: "('c, 'd) cyclic_group_scheme" assumes "cyclic_group G" shows "0 < order G"
map_rbt_greater: fixes k :: "'a"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'd"   and t :: "('a, 'e) rbt" shows "(k \<guillemotleft>| RBT_Impl.map f t) = (k \<guillemotleft>| t)"
intro: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "continuous F"   and "normal_axioms F" shows "normal F"
diff_assign_axiom_def: shows "diff_assign_axiom \<equiv> [[DiffAssign vid1 ($f fid1 local.empty)]]$\<phi> vid1 (singleton ($' vid1)) \<leftrightarrow> $\<phi> vid1 (singleton ($f fid1 local.empty))"
advantage\<^sub>1_def: fixes key_gen :: "nat \<Rightarrow> ('a \<times> 'b) spmf"   and sign :: "nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd spmf"   and verify :: "nat \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and valid_message :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and \<A> :: "nat \<Rightarrow> ('c \<times> 'd, unit + 'c, 'a + 'd) gpv"   and \<eta> :: "nat" shows "suf_cma.advantage\<^sub>1 key_gen sign verify valid_message \<A> \<eta> = spmf (suf_cma.suf_cma\<^sub>1 key_gen sign verify valid_message \<A> \<eta>) True"
cUn_assoc: fixes A :: "'d cset"   and B :: "'d cset"   and C :: "'d cset" shows "cUn (cUn A B) C = cUn A (cUn B C)"
assms: shows "path (g_E (degeneralize_ext m)) u p v"
path_connectedin_iff_path_connected_real: fixes S :: "real set" shows "path_connectedin euclideanreal S = path_connected S"
at_discrete: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and S :: "'b set" assumes "class.discrete_topology open" shows "topological_space.at_within open x S = bot"
selectlike_empty: fixes X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real" shows "selectlike X \<nu> {} = X"
quotient_is_domain: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "primeideal I R" shows "domain (R Quot I)"
nullhomotopic_into_sphere_extension: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and r :: "real" assumes "closed S"   and "continuous_on S f"   and "S \<noteq> {}"   and "f ` S \<subseteq> sphere a r" shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere a r) f (\<lambda>x. c)) = (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> sphere a r \<and> (\<forall>x\<in>S. g x = f x))"
llength_LCons: fixes x :: "'g"   and xs :: "'g llist" shows "llength (LCons x xs) = eSuc (llength xs)"
submonoid_axioms: shows "submonoid (\<eta> ` R) R' (\<cdot>') \<one>'"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "\<And>x y. hom (x * y) = hom x * hom y" shows "times_hom hom"
induct: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" assumes "hindrance_by \<Gamma> f \<epsilon>" obtains a :: "'a" where "a \<in> A \<Gamma>" and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" and "d_OUT f a < weight \<Gamma> a" and "\<epsilon> < weight \<Gamma> a - d_OUT f a"
smap_singleton_mono: fixes v :: "'a set"   and v' :: "'a set"   and k :: "'b" assumes "v \<subseteq> v'" shows "{ k := v}. \<subseteq>. { k := v'}."
arr_lunit: fixes a :: "'c" assumes "C.ide a" shows "C.arr \<l>\<^sub>C[a]"
subst_atm_mset_lists_def: fixes AAs :: "'a multiset list"   and \<sigma>s :: "'s list" shows "AAs \<cdot>\<cdot>aml \<sigma>s = Map2.map2 (\<cdot>am) AAs \<sigma>s"
DOWN_def: fixes t :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "SC.DOWN t \<equiv> the_elem (concrete_category.Dom t)"
induct: fixes P :: "int \<Rightarrow> atom \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "atom" assumes "\<And>m i k ks. P m (Le i (k # ks))"   and "\<And>m d i k ks. P m (Dvd d i (k # ks))"   and "\<And>m d i k ks. P m (NDvd d i (k # ks))"   and "\<And>uu_ v. P uu_ (Le v [])"   and "\<And>uu_ v va. P uu_ (Dvd v va [])"   and "\<And>uu_ v va. P uu_ (NDvd v va [])" shows "P a0 a1"
not_prime_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "\<not> normalization_semidom.prime zero times one normalize one"
R2: shows "gauss_jordan_single A \<in> carrier_mat n nc"
insert_subset_eq_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(add_mset a A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># remove1_mset a B)"
smult_r_distr: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and a :: "'e"   and x :: "'g"   and y :: "'g" assumes "module R M"   and "a \<in> carrier R"   and "x \<in> carrier M"   and "y \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y"
inj_on_image_Pow: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "inj_on ((`) f) (Pow A)"
Polys_closed_monomial: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and c :: "'d" assumes "t \<in> .[X]" shows "monomial c t \<in> P[X]"
cone_decomp_exact: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "cone_decomp T qs" shows "cone_decomp T (exact k qs)"
trivial_outside: fixes v :: "'a" assumes "v \<notin> \<Union> X" shows "g v = v"
adopt_node_locs_impl: shows "adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_adopt_node_locs"
seq_decl_elim: fixes upd :: "var\<^sub>C ModeUpd"   and c :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>(Skip@[upd]) ;; c, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" shows "c' = Stop ;; c \<and> mem' = mem \<and> mds' = C.update_modes upd mds"
simps: fixes a1 :: "nat"   and a2 :: "com" shows "(a1 \<turnstile> a2) = ((\<exists>l. a1 = l \<and> a2 = SKIP) \<or> (\<exists>x a l. a1 = l \<and> a2 = x ::= a \<and> sec a \<le> sec x \<and> l \<le> sec x) \<or> (\<exists>l c\<^sub>1 c\<^sub>2. a1 = l \<and> a2 = c\<^sub>1;; c\<^sub>2 \<and> l \<turnstile> c\<^sub>1 \<and> l \<turnstile> c\<^sub>2) \<or> (\<exists>b l c\<^sub>1 c\<^sub>2. a1 = l \<and> a2 = IF b THEN c\<^sub>1 ELSE c\<^sub>2 \<and> max (sec b) l \<turnstile> c\<^sub>1 \<and> max (sec b) l \<turnstile> c\<^sub>2) \<or> (\<exists>b c. a1 = 0 \<and> a2 = WHILE b DO c \<and> sec b = 0 \<and> 0 \<turnstile> c))"
ins_inorder: fixes t :: "'a btree"   and k :: "nat"   and x :: "'a" assumes "sorted_less (inorder t)" shows "inorder_up\<^sub>i (ins k x t) = ins_list x (inorder t)"
foldr_conv_fold: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and xs :: "'h list" shows "foldr f xs = fold f (rev xs)"
ih_s: assumes "wary s" shows "wt (subst \<rho> s) = wt s + extra_wt \<rho> s"
alist_and_append: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and l1 :: "'c negation_type list"   and l2 :: "'c negation_type list"   and a :: "action"   and p :: "'d" shows "matches \<gamma> (alist_and (l1 @ l2)) a p = matches \<gamma> (MatchAnd (alist_and l1) (alist_and l2)) a p"
max_absorb2: fixes x :: "'d"   and y :: "'d" assumes "x \<le> y" shows "max x y = y"
Basis_list_prod_def: shows "Basis_list = zip Basis_list (replicate (length Basis_list) (0::'b)) @ zip (replicate (length Basis_list) (0::'a)) Basis_list"
3: shows "x\<^sup>T\<^sup>\<star> ; r \<le> x\<^sup>\<star> ; r"
unsigned_word_eqI: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and v :: "'l word"   and w :: "'l word" assumes "class.semiring_char_0 one times plus zero"   and "semiring_1.unsigned one plus zero v = semiring_1.unsigned one plus zero w" shows "v = w"
o_def: fixes f :: "'m \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'm" shows "f \<circ> g = (\<lambda>x. f (g x))"
bounded_UN: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A"   and "\<forall>x\<in>A. bounded (B x)" shows "bounded (\<Union> (B ` A))"
gval_fold_rev_equiv_fold: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gAnd (rev G) (Bc True)) s = gval (fold gAnd G (Bc True)) s"
measure_eq_0_null_sets: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> null_sets M" shows "Sigma_Algebra.measure M S = 0"
Amin_geTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant"   and z :: "ant" shows "(\<forall>j\<le>n. f j \<in> Z\<^sub>\<infinity>) \<and> (\<forall>j\<le>n. z \<le> f j) \<longrightarrow> z \<le> Amin n f"
assms_1: shows "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c"
adjl_\<alpha>_def: fixes l :: "'a list \<times> ('a \<times> 'b \<times> 'a) list" shows "adjl_\<alpha> l \<equiv> let (nl, el) = l in \<lparr>nodes = set nl \<union> fst ` set el \<union> snd ` snd ` set el, edges = set el\<rparr>"
finite_dimensional_vector_space_ow_axioms_def: fixes U\<^sub>V\<^sub>S :: "'d set"   and plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S :: "'d"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set" shows "finite_dimensional_vector_space_ow_axioms U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis \<equiv> (basis \<subseteq> U\<^sub>V\<^sub>S \<and> finite basis) \<and> (with (0::'e) zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>independent\<guillemotright> basis) \<and> (with zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>span\<guillemotright> basis) = U\<^sub>V\<^sub>S"
simps: fixes l1 :: "int option"   and h1 :: "int option"   and l2 :: "int option"   and h2 :: "int option" shows "le_aux (I l1 h1) (I l2 h2) = (le_option False l2 l1 \<and> le_option True h1 h2)"
circuit_nonempty: fixes X :: "'a set" assumes "circuit X" shows "X \<noteq> {}"
mono_mndet_FD: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process"   and P' :: "'b \<Rightarrow> 'b process" assumes "\<forall>x\<in>A. P x \<le> P' x" shows "mndet A P \<le> mndet A P'"
eucl_inf: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "inf x y = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (inf (inner x i) (inner y i)) i) Basis"
exhaust: fixes y :: "('a, 'b, 'c) prog" obtains x :: "'a \<Rightarrow> 'b + 'c \<times> 'a" where "y = Prog x"