hcju/isabelleps huggingface dataset:
axioms_1: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "Limit.cone J C D a \<chi>" shows "category C"
continuous_on_topological: fixes s :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
fns3: fixes fr :: "'b \<Rightarrow> 'b"   and fa :: "'a \<Rightarrow> 'a" assumes "Abs \<circ> fr \<circ> Rep = fa" shows "(norm \<circ> fr \<circ> norm = norm \<circ> fr) = (fa \<circ> Abs = Abs \<circ> fr)"
axioms_2: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "open_cover_of_open_subset X is_open U I C" shows "open_cover_of_open_subset_axioms is_open U"
simps_3: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_sat (IF v t e) = (case ifex_sat e of None \<Rightarrow> case ifex_sat t of None \<Rightarrow> None | Some a \<Rightarrow> Some (a(v := True)) | Some a \<Rightarrow> Some (a(v := False)))"
arr_Rel_comp_Rel_id_Rel_left: fixes \<alpha> :: "V"   and F :: "V"   and A :: "V" assumes "arr_Rel \<alpha> F"   and "F\<lparr>ArrCod\<rparr> = A" shows "id_Rel A \<circ>\<^sub>R\<^sub>e\<^sub>l F = F"
Longest_common_prefix_image_Cons: fixes L :: "'c list set"   and x :: "'c" assumes "L \<noteq> {}" shows "Longest_common_prefix ((#) x ` L) = x # Longest_common_prefix L"
hd_b_in_verts: shows "hd b \<in> set (g_V G)"
lookup_times_monomial_left: fixes t :: "'b"   and c :: "'a"   and p :: "'b \<Rightarrow>\<^sub>0 'a"   and s :: "'b" shows "lookup (monomial c t * p) s = (c * lookup p (s - t) when t adds s)"
Basis_le_norm: fixes b :: "'c"   and x :: "'c" assumes "b \<in> Basis" shows "\<bar>x \<bullet> b\<bar> \<le> norm x"
k_\<Lambda>_PBD_def: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and index :: "int"   and u_block_size :: "int" shows "k_\<Lambda>_PBD point_set block_collection index u_block_size \<equiv> pairwise_balance point_set block_collection index \<and> block_design point_set block_collection u_block_size \<and> k_\<Lambda>_PBD_axioms u_block_size"
rupture_of_Irr: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "field (Rupt K (Irr K x))"
interval_ne_empty_cart_1: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "(cbox a b \<noteq> {}) = (\<forall>i. a $ i \<le> b $ i)"
row_to_poly_vec_sum: fixes ts :: "'t list"   and n :: "nat"   and f :: "'d \<Rightarrow> nat \<Rightarrow> 'c"   and I :: "'d set" assumes "distinct ts"   and "length ts = n" shows "row_to_poly ts (vec n (\<lambda>j. \<Sum>i\<in>I. f i j)) = (\<Sum>i\<in>I. row_to_poly ts (vec n (f i)))"
inv_tR: shows "invertible (Finite_Cartesian_Product.transpose (snd (QR_decomposition A)))"
partitions_of_functions_of: fixes A :: "'c set"   and B :: "'d set"   and P :: "'c set set" assumes "finite A"   and "finite B"   and "partition_on A P"   and "card P \<le> card B" shows "partitions_of A B (functions_of P A B) = P"
known_ptr_not_document_ptr: fixes ptr :: "(_) object_ptr" assumes "\<not> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"   and "DocumentClass.known_ptr ptr" shows "CharacterDataClass.known_ptr ptr"
poly_power_n_eq: fixes n :: "nat"   and a :: "'a"   and x :: "'a" assumes "n \<noteq> 0" shows "(poly ([:- a, 1::'a:] ^ n) x = (0::'a)) = (x = a)"
zero_left: fixes b :: "'b" shows "0 *\<^sub>V b = (0::'a)"
P99: shows "Col P Q B' \<and> P Q Perp B B'"
tm_Max: fixes t :: "real" assumes "t \<in> {s \<in> {t1..<t2}. flow0 x s \<in> {a<--<b}}" shows "t \<le> tm"
greatest_less_ma: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'m) \<le> ma"
Equivalence_def: shows "Equivalence = {(a, b). \<exists>A\<in>P. a \<in> A \<and> b \<in> A}"
iter_def: fixes nrandom :: "natural"   and sz :: "natural"   and seed :: "natural \<times> natural" shows "iter nrandom sz seed = iter' TYPE('b) nrandom sz seed"
ant_minus_inj: fixes z :: "int"   and w :: "int" assumes "ant z \<noteq> ant w" shows "- ant z \<noteq> - ant w"
vimage_algebra_cong: fixes X :: "'c set"   and Y :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and M :: "'d measure"   and N :: "'d measure" assumes "X = Y"   and "\<And>x. x \<in> Y \<Longrightarrow> f x = g x"   and "sets M = sets N" shows "vimage_algebra X f M = vimage_algebra Y g N"
ltl_convert_correct_1: fixes \<phi> :: "expr ltlc"   and APs :: "expr iarray"   and \<phi>\<^sub>i :: "nat ltlc" assumes "ltl_convert \<phi> = (APs, \<phi>\<^sub>i)" shows "atoms_ltlc \<phi> = set (IArray.list_of APs)"
sum_0_all: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "\<forall>a\<in>A. 0 \<le> f a"   and "sum f A = 0"   and "finite A" shows "\<forall>a\<in>A. f a = 0"
sets_sup: fixes A :: "'b measure"   and M :: "'b measure"   and B :: "'b measure" assumes "sets A = sets M"   and "sets B = sets M" shows "sets (sup A B) = sets M"
invar: fixes s :: "('a, unit) RBT.rbt" shows "rs.invar s"
rel_funI: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
LtA_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F \<equiv> Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F \<and> \<not> Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"
assms: fixes B' :: "'a set" assumes "B' \<in> B" shows "open B'"
poly_mapping_size_single: fixes f :: "'b \<Rightarrow> nat"   and g :: "'c \<Rightarrow> nat"   and k :: "'b"   and v :: "'c" shows "poly_mapping_size f g (single k v) = (if v = (0::'c) then g (0::'c) else g (0::'c) + f k + g v + 1)"
distinct_3: fixes x1 :: "'a"   and x3 :: "'a formula" shows "Atom x1 \<noteq> \<^bold>\<not> x3"
strict_trans: fixes ko :: "'b key_order"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "ko.lt ko b a"   and "ko.lt ko c b" shows "ko.lt ko c a"
red_preserves_lconf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and T :: "ty" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "P,E,h \<turnstile> e : T"   and "P,h \<turnstile> l (:\<le>)\<^sub>w E"   and "P \<turnstile> E \<surd>" shows "P,h' \<turnstile> l' (:\<le>)\<^sub>w E"
analz_NI_EI_K_synth_analz_NI_E_K: fixes NI :: "msg set"   and K :: "msg set"   and I :: "msg set"   and Bad :: "agent set" assumes "NI \<subseteq> payload"   and "K \<subseteq> range LtK"   and "I \<subseteq> valid"   and "Keys_bad K Bad" shows "analz (NI \<union> extractable K I \<union> K \<union> Tags) \<subseteq> synth (analz (extr Bad NI (local.abs I) \<union> K \<union> Tags)) \<union> - payload"
suffix_tail: fixes s :: "nat \<Rightarrow> 'a" shows "s |\<^sub>s 1 = tail s"
log2_lm5: fixes u :: "nat" assumes "0 < u" shows "log2 u \<in> nat_to_set u"
R_def: shows "R = (\<lambda>n. EM_remainder' 1 (\<lambda>x. (- 1) ^ 1 * pochhammer s 1 * complex_of_real (x + a) powr (- s - of_nat 1)) (real 0) (real n))"
sup_state_opt_None_any: 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 s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> None \<le>' s"
sturm_ext_R: fixes p :: "real poly" assumes "p \<noteq> 0" shows "int (proots_count p UNIV) = changes_R_smods_ext p (pderiv p)"
RepFun_ecut: fixes y :: "hf"   and z :: "hf"   and f :: "hf \<Rightarrow> hf" assumes "y \<le> z" shows "RepFun y (ecut f z) = RepFun y f"
csegment_midpoint_subset: fixes a :: "'b"   and b :: "'b" shows "closed_segment (midpoint a b) b \<subseteq> closed_segment a b"
right_unique_rel_gcI: assumes "right_unique rel_c" shows "right_unique rel_gc"
fminus_triv: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<inter>| B = {||}" shows "A |-| B = A"
fv_subset: fixes t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "t \<in> M" shows "fv t \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"
size_filter_mset_lesseq: fixes f :: "'a \<Rightarrow> bool"   and M :: "'a multiset" shows "size (filter_mset f M) \<le> size M"
powr_powr: fixes x :: "real"   and a :: "real"   and b :: "real" shows "(x powr a) powr b = x powr (a * b)"
fds_exp_const: fixes c :: "'b" shows "fds_exp (fds_const c) = fds_const (exp c)"
filterlim_compose: fixes g :: "'a \<Rightarrow> 'b"   and F3 :: "'b filter"   and F2 :: "'a filter"   and f :: "'e \<Rightarrow> 'a"   and F1 :: "'e filter" assumes "filterlim g F3 F2"   and "filterlim f F2 F1" shows "LIM x F1. g (f x) :> F3"
Diff_insert0: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A" shows "A - (x \<triangleright> B) = A - B"
diff_union_swap2: fixes y :: "'b"   and M :: "'b multiset"   and x :: "'b" assumes "y \<in># M" shows "add_mset x M - {#y#} = add_mset x (M - {#y#})"
rep_eq: fixes x :: "'k multiset"   and xa :: "'k multiset" shows "count (x + xa) = (\<lambda>a. count x a + count xa a)"
has_integral_vec1_D: fixes f :: "(real, 1) vec \<Rightarrow> 'c"   and y :: "'c"   and S :: "(real, 1) vec set" assumes "(f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)" shows "(f has_integral y) S"
decz: fixes s :: "'a"   and v :: "'b"   and sn :: "'a"   and sz :: "'a"   and M :: "('a \<times> ('a, 'b) Op) set"   and vs :: "'b \<Rightarrow> nat" assumes "(s, Dec v sn sz) \<in> M"   and "vs v = 0" shows "((s, vs), sz, vs) \<in> step M"
abs_eq: shows "(-) = (\<lambda>x y. Abs_atoms (Rep_atoms x - Rep_atoms y))"
validE_def: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool"   and Q :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and E :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>, \<lbrace>E\<rbrace> \<equiv> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>v s. case v of Inl e \<Rightarrow> E e s | Inr r \<Rightarrow> Q r s\<rbrace>"
atUV_def: fixes U :: "'a set"   and V :: "'a set"   and c :: "'a list multiset" shows "atUV U V c \<equiv> \<exists>su sv. {#su#} + {#sv#} \<subseteq># c \<and> atU_s U su \<and> atU_s V sv"
NDC_def: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and c :: "'a" shows "WHATWHERE.NDC SR DA d c \<equiv> \<forall>m m'. WHATWHERE.d_equal DA d m m' \<longrightarrow> WHATWHERE.d_equal DA d (WHATWHERE.NextMem SR c m) (WHATWHERE.NextMem SR c m')"
eq_acom_iff_strip_annos: fixes C1 :: "'a acom"   and C2 :: "'a acom" shows "(C1 = C2) = (strip C1 = strip C2 \<and> annos C1 = annos C2)"
NewArray_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and T :: "ty" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (newA T\<lfloor>e\<rceil>, xs) (newA T\<lfloor>e'\<rceil>, xs')"
face_of_subset_rel_frontier: fixes T :: "'b set"   and S :: "'b set" assumes "T face_of S"   and "T \<noteq> S" shows "T \<subseteq> rel_frontier S"
kruskal_spanning_invariant_def: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "class.stone_kleene_relation_algebra_arc star one times sup less_eq less bot inf top conv uminus" shows "stone_kleene_relation_algebra_arc.kruskal_spanning_invariant star one times less_eq inf conv uminus f g h \<equiv> conv g = g \<and> h = conv h \<and> inf g (uminus (uminus h)) = h \<and> stone_kleene_relation_algebra_arc.spanning_forest star one times less_eq conv uminus f (inf (uminus h) g)"
adds_term_def: fixes u :: "'a \<times> 'b"   and v :: "'a \<times> 'b" shows "pprod.adds_term u v = (pprod.component_of_term u = pprod.component_of_term v \<and> pprod.pp_of_term u adds pprod.pp_of_term v)"
finite_nominals: fixes p :: "('d, 'c) fm" shows "finite (nominals p)"
extend_set_subset_Compl_eq: fixes A :: "'a set"   and B :: "'a set" shows "(extend_set h A \<subseteq> - extend_set h B) = (A \<subseteq> - B)"
Domainp_is_poly: shows "Domainp poly_rel = is_poly ff_ops"
simplex_likeI: fixes P :: "'b set"   and f :: "'b \<Rightarrow> nat set"   and A :: "nat set" assumes "finite P"   and "OrderingSetIso (\<^bold>\<le>*) (\<^bold><*) dual_order.greater_eq dual_order.greater P f"   and "f ` P = Pow A" shows "codomain.simplex_like P"
length_sorted_list_of_set: fixes less_eq :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and less :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and A :: "'q set" assumes "class.linorder less_eq less" shows "length (linorder.sorted_list_of_set less_eq A) = card A"
Red_in_Sup: fixes Ns :: "'f set llist" assumes "chain (\<rhd>) Ns" shows "Sup_llist Ns - Liminf_llist Ns \<subseteq> Red_F (Sup_llist Ns)"
in_hom_char: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.VV.in_hom f a b = (Span.VV.arr a \<and> Span.VV.arr b \<and> Span.VV.arr f \<and> Span.VxV.in_hom f a b)"
Rep_range: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "type_definition Rep Abs A" shows "range Rep = A"
trans_trancl: fixes r :: "('k \<times> 'k) set" shows "trans (r\<^sup>+)"
elims_2: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "'f \<Rightarrow> 'd list" assumes "is_det_state_cover_ass x xa" obtains M :: "('d, 'e, 'f) FSM" and f :: "'f \<Rightarrow> 'd list" where "x = M" and "xa = f" and "f (initial M) = [] \<and> (\<forall>s\<in>d_reachable M (initial M). d_reaches M (initial M) (f s) s)"
decision: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and X :: "final_decision" shows "\<gamma>,p\<turnstile> \<langle>rs, Decision X\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision X"
span_is_subset2: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "span S \<subseteq> carrier V"
of_int: fixes n :: "int" shows "f (of_int n) = f (0::'a)"
OO_eq: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "R OO (=) = R"
R_nonneg: fixes x :: "'a"   and y :: "'a" shows "0 \<le> R x y"
rel_fun_eq_onp_rel: fixes R :: "'c \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (eq_onp R) S = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
nat_of_integer_non_positive: fixes k :: "integer" assumes "k \<le> 0" shows "nat_of_integer k = 0"
sat_eq: fixes c :: "('d, 'e) conf" shows "Conf.sat c = (states c \<noteq> {})"
Suc_less_eq2: fixes n :: "nat"   and m :: "nat" shows "(Suc n < m) = (\<exists>m'. m = Suc m' \<and> n < m')"
f12: shows "uint (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) * uint (get_operand2 (snd instr) s2) = uint (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) * uint (get_operand2 (snd instr) s1)"
assms_2: shows "(f ^^ Suc k) a = (f ^^ k) a"
cogen_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and cs :: "'c list"   and w :: "'d" shows "cogen P cs w \<equiv> \<exists>c\<in>set cs. P c w"
t_prod_iff: fixes w :: "'a \<times> ('b \<times> 'a) list"   and X :: "('a \<times> ('b \<times> 'a) list) set"   and Y :: "('a \<times> ('b \<times> 'a) list) set" shows "(w \<in> t_prod X Y) = (\<exists>u v. w = t_fusion u v \<and> u \<in> X \<and> v \<in> Y \<and> Dioid_Models_New.last u = first v)"
that: shows "3 * sqrt (x * x + 1 / 3 - x) < sqrt 3"
project_Stable_I: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set" assumes "extend h F \<squnion> G \<in> Stable (extend_set h A)" shows "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Stable A"
unique_LID_M2: fixes M :: "agent"   and LID_M :: "nat"   and Trans :: "msg"   and evs :: "event list"   and Trans' :: "msg" assumes "Notes M \<lbrace>Number LID_M, Trans\<rbrace> \<in> set evs"   and "Notes M \<lbrace>Number LID_M, Trans'\<rbrace> \<in> set evs"   and "evs \<in> set_pur" shows "Trans' = Trans"
no_loop: fixes x :: "'v" shows "\<not> edge \<Gamma> x x"
llist_of_list_of: fixes xs :: "'c llist" assumes "lfinite xs" shows "llist_of (list_of xs) = xs"
trans_approx: fixes x :: "'a pmf"   and y :: "'a pmf"   and z :: "'a pmf" assumes "x \<approx>[\<R>] y"   and "y \<approx>[\<R>] z" shows "x \<approx>[\<R>] z"
normalizeE: fixes x :: "trm"   and xa :: "trm" assumes "pred.eval (Execute.normalize x) xa" obtains "x \<Down> xa"
hyps_2: fixes mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mds\<^sub>2' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "mds\<^sub>2 \<sqsubseteq> mds\<^sub>2''" shows "\<forall>mem. (\<langle>Stop, mds\<^sub>2', mem'\<rangle> \<in> loc_reach \<langle>c, mds\<^sub>2, mem\<rangle> \<longrightarrow> mds\<^sub>2' \<sqsubseteq> mds\<^sub>2'''__) \<and> locally_sound_mode_use \<langle>c, mds\<^sub>2, mem\<rangle>"
scan_append: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list"   and a :: "'b" shows "scan f (xs @ ys) a = scan f xs a @ scan f ys (fold f xs a)"
add_diff_cancel_right': fixes a :: "'b"   and b :: "'b" shows "a + b - b = a"
leg_simps_3: shows "C.arr HH\<nu>\<pi>\<rho>.leg1"
map_pair_id: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "map_pair id xs = xs"
degree_diff_le: fixes p :: "'a poly"   and n :: "nat"   and q :: "'a poly" assumes "degree p \<le> n"   and "degree q \<le> n" shows "degree (p - q) \<le> n"
ipurge_fail_aux_t_subset: fixes Z :: "('b, 'c) ipurge_rec"   and Y :: "('b, 'c) ipurge_rec" assumes "Z \<in> ipurge_fail_aux_t_set Y" shows "ipurge_fail_aux_t_set Z \<subseteq> ipurge_fail_aux_t_set Y"
insert_absorb: fixes a :: "'x"   and A :: "'x set" assumes "a \<in> A" shows "insert a A = A"
delete_edge_def: fixes v :: "'a"   and v' :: "'a"   and G :: "('a, 'b) graph_scheme" shows "delete_edge v v' G \<equiv> \<lparr>nodes = nodes G, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> (e1, e2) \<noteq> (v, v')}\<rparr>"
simps_1: shows "nat_of_num num.One = Suc 0"
Pos_atm_of_iff: fixes L :: "'a literal" shows "(Pos (atm_of L) = L) = is_pos L"
prod_ord_rawI: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and ys :: "('b \<times> 'd) list \<times> 'a"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "oalist_raw.oalist_inv rep_key_order xs"   and "oalist_raw.oalist_inv rep_key_order ys"   and "\<And>k. k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys) \<Longrightarrow> P k (oalist_raw.lookup_raw rep_key_order xs k) (oalist_raw.lookup_raw rep_key_order ys k)" shows "oalist_raw.prod_ord_raw rep_key_order P xs ys"
assms_2: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (G i)"
conv_complement_sub: fixes x :: "'a"   and y :: "'a" shows "id x \<sqinter> - (x \<sqinter> y) \<squnion> - y = - y"
emeasure_additive: fixes M :: "'b measure" shows "Measure_Space.additive (sets M) (emeasure M)"
those_eq_Some_map_Some_iff: fixes xs :: "'a option list"   and ys :: "'a list" shows "(those xs = Some ys) = (xs = map Some ys)"
lookup_update: fixes k :: "'d"   and v :: "'c"   and m :: "('d, 'c) mapping" shows "Mapping.lookup (Mapping.update k v m) k = Some v"
bezout_coefficients: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a" assumes "bezout_coefficients a b = (x, y)" shows "x * a + y * b = gcd a b"
BNTr1: fixes a :: "'a" assumes "a \<in> carrier D" shows "Worder (Iod D {a})"
simps_2: fixes i :: "nat"   and x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set" shows "rel_chain_fun (Suc i) x y r = (if i = 0 then y else SOME z. (rel_chain_fun i x y r, z) \<in> r)"
target: fixes T :: "'b"   and Enc :: "'a \<Rightarrow> 'b" shows "(TargetTerm T, TargetTerm T) \<in> encoding.indRelRPO Enc"
repeat_stabilize_process_call: fixes \<Gamma> :: "(char list \<times> 'c rule list) list"   and chain_name :: "char list"   and default_action :: "action"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and n :: "nat"   and s :: "state"   and t :: "state" assumes "sanity_wf_ruleset \<Gamma>"   and "chain_name \<in> set (map fst \<Gamma>)"   and "default_action = Accept \<or> default_action = Drop" shows "map_of \<Gamma>,\<gamma>,p\<turnstile> \<langle>repeat_stabilize n (process_call (map_of \<Gamma>)) [Rule MatchAny (Call chain_name), Rule MatchAny default_action], s\<rangle> \<Rightarrow> t = map_of \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule MatchAny (Call chain_name), Rule MatchAny default_action], s\<rangle> \<Rightarrow> t"
RM_subset: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and i :: "nat" shows "RM M2 M1 \<Omega> V m i \<subseteq> C M2 M1 \<Omega> V m i"
one_le_numeral: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one (numeral.numeral one plus n)"
is_lubI: fixes S :: "'a set"   and x :: "'a" assumes "S <| x"   and "\<And>u. S <| u \<Longrightarrow> x \<sqsubseteq> u" shows "S <<| x"
idem: fixes a :: "'d multiset" shows "a \<inter># a = a"
scaleRat_leq1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and k :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq a b"   and "0 < k" shows "less_eq (scaleRat k a) (scaleRat k b)"
has_bochner_integral_norm_bound: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "has_bochner_integral M f x" shows "ennreal (norm x) \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"