hcju/isabelleps huggingface dataset:
bind_is_OK_E2: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and x :: "'d" assumes "h \<turnstile> ok (f \<bind> g)"   and "h \<turnstile> f \<rightarrow>\<^sub>r x" obtains h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> ok g x"
Ri_indep: fixes N' :: "'a literal multiset set"   and N :: "'a literal multiset set" assumes "N' \<subseteq> sr.Rf N" shows "sr.Ri N \<subseteq> sr.Ri (N - N')"
strictly_subsumes_def: fixes C :: "('c, 'd) Term.term literal multiset"   and D :: "('c, 'd) Term.term literal multiset" shows "strictly_subsumes C D = strict subsumes C D"
Ostrowski_first: fixes vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc 0) vv" shows "\<exists>x\<in>carrier K. Ostrowski_elem K (Suc 0) vv x"
gt_sym_trans: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a"   and g :: "'a"   and f :: "'a" assumes "gt_sym gt_sym"   and "gt_sym h g"   and "gt_sym g f" shows "gt_sym h f"
incrIndexList_1len: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "Suc 0 < nmax"
mem_aforms_err_Cons_iff: fixes x :: "real"   and xs :: "real list"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and XS :: "((real \<times> real pdevs) \<times> real) list" shows "(x # xs \<in> aforms_err e (X # XS)) = (x \<in> aform_err e X \<and> xs \<in> aforms_err e XS)"
prems_4: shows "((X\<^sub>r \<mu> \<in> old (fst x) \<longrightarrow> \<mu> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> \<mu> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>nd\<in>snd x. \<forall>nd'\<in>snd x. X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> \<mu> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)"
ring_iso_memE_2: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "h \<in> ring_iso R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
exhaust_sel: fixes labeled_graph :: "('e, 'f) labeled_graph" obtains "labeled_graph = LG (edges labeled_graph) (vertices labeled_graph)"
vc_complete: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}" shows "\<exists>C. Nielson_VCG.strip C = c \<and> Nielson_VCG.vc C Q \<and> (\<forall>l s. P l s \<longrightarrow> Nielson_VCG.pre C Q l s \<and> Q l (Nielson_VCG.postQ C s)) \<and> (\<exists>k. \<forall>l s. P l s \<longrightarrow> Nielson_VCG.time C s \<le> k * e s)"
cs_recording_implies_snapshot: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "snd (cs (S t i) cid) = Recording"   and "channel cid = Some (p, q)" shows "ps (S t i) q \<noteq> None"
k2_var: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> z \<le> y"
content_split: fixes k :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "real" assumes "k \<in> Basis" shows "content (cbox a b) = content (cbox a b \<inter> {x. x \<bullet> k \<le> c}) + content (cbox a b \<inter> {x. c \<le> x \<bullet> k})"
itrev: fixes xs :: "'a list"   and ys :: "'a list" shows "itrev xs ys = rev xs @ ys"
finite_acyclic_wf_converse: fixes r :: "('c \<times> 'c) set" assumes "finite r"   and "acyclic r" shows "wf (r\<inverse>)"
coeff_mult_degree_sum: fixes p :: "'a poly"   and q :: "'a poly" shows "poly.coeff (p * q) (degree p + degree q) = lead_coeff p * lead_coeff q"
matrix_vector_mul_1: fixes g :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) g" shows "(*v) (matrix g) = g"
unsynlr: fixes x :: "('a, 'b) synlr" shows "unsynlr x \<in> {x. x \<in> synlr}"
simps: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec" shows "maximize A b c = (if dim_vec b = dim_row A \<and> dim_vec c = dim_col A then Some (optimize_no_cond A b c) else None)"
it_dfs_correct: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization" assumes "DFS G param" shows "param_DFS_defs.it_dfs G param \<le> SPEC (\<lambda>s. DFS_invar G param s \<and> \<not> param_DFS_defs.cond G param s)"
sint_range: fixes ra :: "'a Word.word" assumes "real_of_int (sint ra) < real_of_int (sint POS_INF)"   and "real_of_int (sint NEG_INF) < real_of_int (sint ra)" shows "sint ra \<in> {i. - (2 ^ 31 - 1) < i \<and> i < 2 ^ 31 - 1}"
RangeE: fixes b :: "'b"   and r :: "('c \<times> 'b) set" assumes "b \<in> Range r" obtains a :: "'c" where "(a, b) \<in> r"
pr_in_hom_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "\<guillemotleft>pr0 a b : HF'.prod a b \<rightarrow> b\<guillemotright>"
map_id: fixes p :: "'b spmf" shows "map_spmf id p = p"
abs_eq: fixes xa :: "('b \<times> bool) list"   and x :: "('b \<times> bool) list" assumes "eq_onp proper_signed_list xa xa"   and "eq_onp proper_signed_list x x" shows "Abs_freeword xa + Abs_freeword x = Abs_freeword (prappend_signed_list xa x)"
RECT_rule: fixes body :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and V :: "('b \<times> 'b) set"   and pre :: "'b \<Rightarrow> bool"   and x :: "'b"   and M :: "'b \<Rightarrow> 'c" assumes "trimono body"   and "wf V"   and "pre x"   and "\<And>f x. \<lbrakk>\<And>x'. \<lbrakk>pre x'; (x', x) \<in> V\<rbrakk> \<Longrightarrow> f x' \<le> M x'; pre x; REC\<^sub>T body = f\<rbrakk> \<Longrightarrow> body f x \<le> M x" shows "REC\<^sub>T body x \<le> M x"
hyps: fixes \<O> :: "nat set"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option" assumes "\<forall>a\<in>\<O> \<union> all_acquired sb. \<S>' a = \<S> a"   and "weak_sharing_consistent \<O> sb" shows "release sb (dom \<S>') \<R> = release sb (dom \<S>) \<R>"
set_mset_union: fixes M :: "'k multiset"   and N :: "'k multiset" shows "set_mset (M + N) = set_mset M \<union> set_mset N"
numeral_Bit0_div_2: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and n :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 num.One)) = numeral.numeral one plus n"
left_tree_initial': fixes G :: "('c, 'd) Graph_scheme"   and s :: "'c"   and t :: "'c" assumes "Tree G"   and "s \<in> V\<^bsub>G\<^esub>"   and "t \<in> V\<^bsub>G\<^esub>"   and "s \<noteq> t" shows "t \<notin> Tree.left_tree G s t"
tau_zero: shows "\<tau> (0::'a) = (0::'a)"
esssup_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" shows "esssup M f = (if f \<in> borel_measurable M then Limsup (ae_filter M) f else \<top>)"
partUnfold: fixes P :: "pi"   and Q :: "pi"   and s :: "(name \<times> name) list" assumes "P \<approx>\<^sup>s Q" shows "P[
] \<approx>\<^sup>s Q[
]"
fsafe: shows "fsafe ([[x := \<theta>]]\<phi>)"
elims_2: fixes x :: "'b uprod set"   and xa :: "'b"   and xb :: "'b uprod list"   and xc :: "'b" assumes "epath x xa xb xc" obtains E :: "'b uprod set" and u :: "'b" and v :: "'b" where "x = E__" and "xa = u__" and "xb = []" and "xc = v__" and "u__ = v__"   | E :: "'b uprod set" and u :: "'b" and xd :: "'b uprod" and xs :: "'b uprod list" and v :: "'b" where "x = E" and "xa = u" and "xb = xd # xs" and "xc = v" and "(\<exists>w. u \<noteq> w \<and> Upair u w = xd \<and> epath E w xs v) \<and> xd \<in> E"
x'_min: shows "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
simple_ruleset_normalize_rules: fixes rs :: "'b rule list"   and f :: "'b match_expr \<Rightarrow> 'b match_expr list" assumes "simple_ruleset rs" shows "simple_ruleset (normalize_rules f rs)"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "a \<le> max a b"
case_prod_conv: fixes f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and a :: "'l"   and b :: "'m" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
empty_def: shows "Mapping.empty \<equiv> mapping.Mapping Map.empty"
sum_var_list_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('c \<times> nat) list"   and x :: "'c" assumes "class.linorder less_eq less" shows "linorder.sum_var_list m x \<equiv> \<Sum>(y, c)\<leftarrow>m. if x = y then c else 0"
polyexp_ratfps: fixes xs :: "('b \<times> nat \<times> 'b) list" shows "fps_of_ratfps (polyexp_ratfps xs) = polyexp_fps xs"
shEx_cong: fixes P :: "'b \<Rightarrow> 'c upred"   and Q :: "'b \<Rightarrow> 'c upred" assumes "\<And>x. P x = Q x" shows "ushEx P = ushEx Q"
emb_step_under_args_num_args: fixes p :: "dir list"   and d :: "dir"   and t :: "('c, 'd) tm" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p" shows "num_args (emb_step_at p d t) = num_args t"
max: fixes a :: "complex" shows "cmod (if 1 < cmod a then cnj a else 1) = max 1 (cmod a)"
supp_of_finite_insert: fixes S :: "'a set"   and x :: "'a" assumes "finite S" shows "supp (insert x S) = supp x \<union> supp S"
add_le_same_cancel1: fixes b :: "'k multiset"   and a :: "'k multiset" shows "(b + a \<subseteq># b) = (a \<subseteq># {#})"
elims_3: fixes x :: "'b heap" assumes "\<not> no_Empty x" obtains "x = heap.Empty"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "\<not> no_Emptys hs"
\<phi>T_def: shows "\<phi>T \<equiv> diag (neg T)"
Infinitesimal_inverse_HFinite: fixes x :: "'a star" assumes "x \<notin> Infinitesimal" shows "inverse x \<in> HFinite"
update_simps_20: fixes T :: "(nat \<times> linear_poly) list"   and s :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>l (\<T>_update T s) = \<B>\<^sub>i\<^sub>l s"
append_eq_appendI: fixes xs :: "'k list"   and xs1 :: "'k list"   and zs :: "'k list"   and ys :: "'k list"   and us :: "'k list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
PartialOrder_def: shows "PartialOrder \<equiv> {P. refl_on (pset P) (order P) \<and> antisym (order P) \<and> trans (order P)}"
term_pp_rel_trans_eq_left: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a"   and b :: "'t \<times> 'a"   and c :: "'t \<times> 'a" assumes "ord_term_lin.is_le_rel r"   and "term_pp_rel (=) a b"   and "term_pp_rel r b c" shows "term_pp_rel r a c"
diffsimps_2: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A - (B \<union> C) = (A - B) \<inter> (A - C)"
span_scale: fixes x :: "'c"   and S :: "'c set"   and c :: "'c" assumes "x \<in> ideal S" shows "c * x \<in> ideal S"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>c n p. (\<And>x. n \<le> x \<Longrightarrow> P p) \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (v)\<^sub>p"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
subsequence_shift: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'a"   and j :: "nat"   and k :: "nat" shows "suffix i w [j \<rightarrow> k] = w [i + j \<rightarrow> i + k]"
cobounded2: fixes b :: "'a Word.word"   and a :: "'a Word.word" shows "b \<le>s signed.max a b"
simps_2: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" shows "is_ord (MKT n l r h) = ((\<forall>n'\<in>set_of l. n' < n) \<and> (\<forall>n'\<in>set_of r. n < n') \<and> is_ord l \<and> is_ord r)"
injective_closed_imp_embedding_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "closed_map X Y f"   and "inj_on f (topspace X)" shows "embedding_map X Y f"
wg: shows "web_flow (quotient_web \<Gamma> f) g"
equals_cod_if_value_is_ide: assumes "C.ide a" shows "A.map = A.G.map"
lookup_def: fixes x :: "'v" shows "lookup x = get (\<lambda>s. case s x of None \<Rightarrow> fail | Some y \<Rightarrow> return y)"
antipar_1: shows "S.trg (S.UP g) = S.src (S.UP f)"
exact_seq_end_iff: fixes G :: "'a monoid"   and H :: "'a monoid"   and g :: "'a \<Rightarrow> 'a"   and q :: "('a \<Rightarrow> 'a) list" shows "exact_seq ([G, H], g # q) = (q = [] \<and> group_hom H G g)"
WT_converter_id: fixes \<I> :: "('i, 'j) \<I>" shows "\<I>, \<I> \<turnstile>\<^sub>C 1\<^sub>C \<surd>"
pow_prop: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "pow_cond t n (PoW t n)"
mask_eq_exp_minus_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask n = minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one"
vec_0: shows "Finite_Cartesian_Product.vec (0::'b) = 0"
distinct_disc_10: fixes event :: "('a, 'b, 'c) event" assumes "isRecv event" shows "\<not> isSend event"
simps_87: fixes f1 :: "'a"   and f2 :: "'b basic_type \<Rightarrow> 'a"   and f3 :: "'b basic_type \<Rightarrow> 'a"   and f4 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f5 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f6 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f7 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f8 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f9 :: "String.literal \<rightharpoonup>\<^sub>f 'b OCL_Types.type \<Rightarrow> 'a"   and x7 :: "'b OCL_Types.type" shows "(case Bag x7 of OclSuper \<Rightarrow> f1 | x[1] \<Rightarrow> f2 x | x[?] \<Rightarrow> f3 x | Collection x \<Rightarrow> f4 x | Set x \<Rightarrow> f5 x | OrderedSet x \<Rightarrow> f6 x | Bag x \<Rightarrow> f7 x | Sequence x \<Rightarrow> f8 x | Tuple x \<Rightarrow> f9 x) = f7 x7"
blinfun_of_cblinfun_cblinfun_compose: fixes f :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and g :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "blinfun_of_cblinfun (f o\<^sub>C\<^sub>L g) = blinfun_of_cblinfun f o\<^sub>L blinfun_of_cblinfun g"
Rep_dense_inverse: fixes x :: "'c dense" shows "Abs_dense (Rep_dense x) = x"
intro: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "comm_monoid_list f z"   and "comm_monoid_set f z" shows "comm_monoid_list_set f z"
iso_Sup_supdistl: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "Sup \<circ> (`) f \<le> f \<circ> Sup"
set_cases: fixes e :: "'e"   and a :: "'e list" assumes "e \<in> set a" obtains z2 :: "'e list" where "a = e # z2__"   | z1 :: "'e" and z2 :: "'e list" where "a = z1 # z2" and "e \<in> set z2"
Act\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and f :: "'d"   and \<alpha> :: "'c"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "p \<bullet> Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> (p \<bullet> f) (p \<bullet> \<alpha>) (p \<bullet> t\<^sub>\<alpha>)"
conv_complement_sub_leq: fixes x :: "'a"   and y :: "'a" shows "id x \<sqinter> - (x \<sqinter> y) \<le> - y"
monotone2monotone: fixes ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and ordc :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and t :: "'h \<Rightarrow> 'i" assumes "\<And>x. monotone ordb ordc (f x)"   and "monotone orda ordb t"   and "\<And>y. monotone orda ordc (\<lambda>x. f x y)"   and "transp ordc" shows "monotone orda ordc (\<lambda>x. f x (t x))"
no_bintr_alt1: fixes n :: "nat" shows "take_bit n = (\<lambda>w. w mod 2 ^ n)"
int_pow_inv: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier R" shows "[i] \<cdot> (\<ominus> x) = \<ominus> ([i] \<cdot> x)"
pls_cong: fixes H :: "fmla set"   and t :: "trm"   and t' :: "trm"   and u :: "trm"   and u' :: "trm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> pls t u EQ pls t' u'"
assms_2: shows "nprv (insert \<phi> F) \<psi>"
S: fixes a :: "nat \<Rightarrow> real"   and b :: "nat \<Rightarrow> real"   and u :: "real" assumes "a \<in> S"   and "b \<in> T"   and "0 < u"   and "u < 1" shows "(\<lambda>i. (1 - u) * a i + u * b i) \<in> S"
drest_smap_drest_smap_drest: fixes vs :: "'c set"   and x :: "('c, 'd) fmap"   and s :: "('c, 'd) fmap" shows "(fmrestrict_set vs x \<subseteq>\<^sub>f s) = (fmrestrict_set vs x \<subseteq>\<^sub>f fmrestrict_set vs s)"
unit_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "horizontal_homs.obj V src a" shows "partial_magma.arr V (\<i> a)"
rtranclp_eq_rtranclp': fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<lambda>x y. P x y \<and> x \<noteq> y)\<^sup>*\<^sup>* = P\<^sup>*\<^sup>*"
bounded: fixes x :: "'a" assumes "x \<in> A" shows "bounded (range (\<lambda>n. f n x))"
dres_order_simps_7: shows "dFAIL \<noteq> dSUCCEED"
fiter0: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> \<sqsubseteq> nil"
fTo_conditional_min1: fixes C :: "'a set"   and Ca :: "'a set set" assumes "C \<in> carrier fTo D"   and "Chain (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca"   and "Ca \<noteq> {}" shows "minimum_elem (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) (upper_bounds (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca) (\<Union> Ca)"
octo_of_real_continuous_on: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "continuous_on S f" shows "continuous_on S (\<lambda>x. octo_of_real (f x))"
simps_6: fixes \<alpha> :: "('e, 'f, 'd) hp"   and \<beta> :: "('e, 'f, 'd) hp" shows "FVP (\<alpha> \<union>\<union> \<beta>) = FVP \<alpha> \<union> FVP \<beta>"
app_list: fixes t1 :: "('a, 'b) FingerTreeStruc"   and t2 :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.toList (FingerTreeStruc.app t1 t2) = FingerTreeStruc.toList t1 @ FingerTreeStruc.toList t2"
FreeR_def: fixes x2 :: "freeRes"   and x1 :: "pi" shows "x2 \<prec> x1 \<equiv> Abs_residual (FreeR_Rep x2 x1)"
ipurge_tr_rev_ipurge_tr_sources_1: fixes D :: "'d \<Rightarrow> 'c"   and y :: "'d"   and I :: "('c \<times> 'c) set"   and u :: "'c"   and zs :: "'d list"   and xs :: "'d list" assumes "D y \<notin> sources I D u (y # zs)" shows "ipurge_tr_rev I D u (xs @ y # zs) = ipurge_tr_rev I D u (xs @ ipurge_tr I D (D y) zs)"
G: shows "foldl (+) 0 ns' = n - Suc (Suc m) \<and> length xs' = n - Suc (Suc m)"
rel_fun_eq_pcr_ennreal: fixes f :: "'c \<Rightarrow> ereal"   and g :: "'c \<Rightarrow> ennreal" shows "rel_fun (=) pcr_ennreal f g = (f = enn2ereal \<circ> g)"
atMost_nat_nonempty: fixes k :: "nat" shows "{..k} \<noteq> {}"
for_all_Suc: fixes P :: "nat \<Rightarrow> bool"   and i :: "nat" assumes "P i" shows "(\<forall>j\<ge>Suc i. P j) = (\<forall>j\<ge>i. P j)"
eventually_at_top_within_linorder: fixes s :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "s \<noteq> {}" shows "eventually P (at_top_within s) = (\<exists>x0\<in>s. \<forall>x\<ge>x0. x \<in> s \<longrightarrow> P x)"
simps: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "'b prog" shows "P \<turnstile> a1 \<prec>\<^sup>1 a2 = (\<exists>C D rest. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(D, rest)\<rfloor> \<and> C \<noteq> Object)"
f_fv: shows "fv_trm f \<subseteq> fv \<phi>"
transpose_transpose: fixes xs :: "'b list list" assumes "sorted (rev (map length xs))" shows "List.transpose (List.transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs"
open_segment_idem: fixes a :: "'b" shows "open_segment a a = {}"
bind_SE_def: fixes f :: "'a \<Rightarrow> ('c \<times> 'a) option"   and g :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'a) option" shows "bind\<^sub>S\<^sub>E f g = (\<lambda>\<sigma>. case f \<sigma> of None \<Rightarrow> None | Some (out, \<sigma>') \<Rightarrow> g out \<sigma>')"
Gauss_Jordan_PA_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "Gauss_Jordan_PA A = Gauss_Jordan_upt_k_PA A (ncols A - 1)"
nn_integral_snd: fixes f :: "'b \<times> 'c \<times> 'd \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>S \<partial>T \<Otimes>\<^sub>M P = integral\<^sup>N (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) f"
index_size_conv: fixes xs :: "'a list"   and x :: "'a" shows "(index xs x = length xs) = (x \<notin> set xs)"
"1_1": shows "u @ v = w\<^sub>1 @ [a] @ w\<^sub>2"
asym: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subset># b"   and "b \<subset># a" shows "False"
int_simps_32: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<not> Q \<longrightarrow> \<not> P) = (P \<longrightarrow> Q)"
moebius_sum_tail_bigo: shows "(\<lambda>x. \<Sum>n. moebius_mu (n + Suc (nat \<lfloor>x\<rfloor>)) / real ((n + Suc (nat \<lfloor>x\<rfloor>))\<^sup>2)) \<in> O((/) 1)"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "map_vertex f (Vertex x1) = Vertex (f x1)"
IH: fixes n :: "'node" assumes "upd_cs cs' as = []"   and "n -as\<rightarrow>* n'"   and "\<forall>a\<in>set cs'. valid_edge a" shows "(if cs' = [] then get_proc n else get_proc (last (sourcenodes cs'))) = get_proc n'"
1: shows "\<forall>b\<in>Y. g b \<in> A \<and> f (g b) = b"
parent_two_children: fixes c :: "int"   and l :: "int"   and p :: "int" shows "(c = l_child l p \<or> c = r_child l p) = (parent l c = p)"
const_bcontfun: fixes b :: "'d" shows "(\<lambda>x. b) \<in> bcontfun"
fixpt_induct: fixes P :: "nat bdd list \<times> bool list \<Rightarrow> bool list list \<Rightarrow> bool"   and M :: "nat bdd list \<times> bool list"   and T :: "bool list list" assumes "\<And>M T. (fst (iter M T) \<Longrightarrow> P M (snd (iter M T))) \<Longrightarrow> P M T" shows "P M T"
IH_2: assumes "Vagree \<nu> \<nu>' (Inl ` FVO ODE2)"   and "Iagree I J ({Inl x |x. Inl x \<in> SIGO ODE2} \<union> {Inr (Inr x) |x. Inr x \<in> SIGO ODE2})" shows "ODE_sem I ODE2 (fst \<nu>) = ODE_sem J ODE2 (fst \<nu>')"
abs_eq: fixes x :: "'d" shows "norm (to_conjugate_space x) = norm x"