hcju/isabelleps huggingface dataset:
fps_mult_of_nat_nth_1: fixes k :: "nat"   and f :: "'a fps"   and n :: "nat" shows "fps_nth (of_nat k * f) n = of_nat k * fps_nth f n"
fmapU_maybe_def: shows "fmapU \<equiv> maybe_map"
Sbis_matchC_C: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>s d" shows "matchC_C Sbis c d"
atU_single: fixes U :: "'a set"   and s :: "'a list" shows "atU U {#s#} = atU_s U s"
lrev_body2_lrev_body1_eq: shows "lrev_body2 = lrev_body1"
divide_strict_left_mono: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less b a"   and "less zero c"   and "less zero (times a b)" shows "less (divide c a) (divide c b)"
id_1: shows "M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gs ! i) [0..<Suc k]) = M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gs ! i) [0..<k]) + g k \<cdot>\<^sub>v gs ! k"
nonsat_push_precond_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "nonsat_push_precond f l \<equiv> \<lambda>(u, v). (0::'a) < excess f u \<and> excess f u < cf_of f (u, v) \<and> (u, v) \<in> cfE_of f \<and> l u = l v + 1"
similar_mat_witD_3: fixes n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "A = P * B * Q"
is_zero_iarray_def: fixes A :: "'b iarray" shows "is_zero_iarray A = IArray.all (\<lambda>i. A !! i = (0::'b)) (IArray [0..<IArray.length A])"
supset_has_bottom_pseudominimal_in_below_in: fixes P :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "dual_order.has_bottom P"   and "x \<in> P"   and "dual_order.pseudominimal_in (dual_order.below_in P x) y" shows "dual_order.pseudominimal_in P y"
clamp_bounded: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and b :: "'d" assumes "bounded (f ` cbox a b)" shows "bounded (range (\<lambda>x. f (clamp a b x)))"
strict_pref_def: fixes x :: "'b"   and r :: "('b \<times> 'b) set"   and y :: "'b" shows "x \<^bsub>r\<^esub>\<prec> y \<equiv> x \<^bsub>r\<^esub>\<preceq> y \<and> (y, x) \<notin> r"
ensures_def: fixes A :: "'b set"   and B :: "'b set" shows "A ensures B \<equiv> (A - B co A \<union> B) \<inter> transient (A - B)"
eq_equivp: shows "equivp (\<sim>\<^sub>L)"
wf_induct_rule: fixes r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "wf r"   and "\<And>x. (\<And>y. (y, x) \<in> r \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
stutter_extend_ex_def: fixes ex :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" shows "stutter_extend_ex ex \<equiv> \<lambda>x. case x of None \<Rightarrow> id | Some a \<Rightarrow> ex a"
fls_shift_subdegree: fixes f :: "'a fls"   and n :: "int" assumes "f \<noteq> 0" shows "fls_subdegree (fls_shift n f) = fls_subdegree f - n"
mask_and_mask: fixes a :: "nat"   and b :: "nat" shows "mask a AND mask b = mask (min a b)"
isShortestPath_alt: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "cf.isShortestPath u p v = (cf.isSimplePath u p v \<and> length p = cf.min_dist u v)"
rewriteL_comp_comp2: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and l1 :: "'d \<Rightarrow> 'b"   and l2 :: "'a \<Rightarrow> 'd"   and h :: "'e \<Rightarrow> 'a"   and r :: "'e \<Rightarrow> 'd" assumes "f \<circ> g = l1 \<circ> l2"   and "l2 \<circ> h = r" shows "f \<circ> (g \<circ> h) = l1 \<circ> r"
cut_less_card: fixes I :: "'b set"   and t :: "'b" assumes "finite I" shows "card (I \<down>< t) \<le> card I"
list_of_vec_map: fixes xs :: "'a vec" shows "list_of_vec xs = map (($) xs) [0..<dim_vec xs]"
pdevs_apply_pdevs_of_ivl: fixes l :: "'b"   and u :: "'b"   and i :: "nat" shows "pdevs_apply (pdevs_of_ivl l u) i = (if i < length Basis_list then ((u - l) \<bullet> Basis_list ! i / 2) *\<^sub>R Basis_list ! i else (0::'b))"
Sim: fixes \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* \<Psi>"   and "distinct (bn \<alpha>)"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C" shows "\<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel"
measurable_P_restriction: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure"   and A :: "'b set" assumes "Measurable.pred M P"   and "A \<in> sets M" shows "{x \<in> A. P x} \<in> sets M"
ideal_generated_mult_unit: fixes S :: "'a set"   and u :: "'a" assumes "finite S"   and "u dvd (1::'a)" shows "ideal_generated ((*) u ` S) = ideal_generated S"
SetPC_split: fixes p :: "'c \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> real"   and f :: "'b \<Rightarrow> bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and ab :: "bool"   and Q :: "'c \<Rightarrow> real" assumes "\<And>x s. x \<in> supp (p s) \<Longrightarrow> P x \<tturnstile> f x ab Q"   and "\<And>s. nneg (p s)" shows "\<lambda>s. \<Sum>x\<in>supp (p s). p s x * P x s \<tturnstile> SetPC f p ab Q"
host_c: shows "h \<turnstile> get_dom_component (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host) \<rightarrow>\<^sub>r host_c"
split_sels_2: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (\<not> (nat = 0 \<and> \<not> P f1 \<or> nat = Suc (nat.pred nat) \<and> \<not> P (f2 (nat.pred nat))))"
case_cong: fixes generat :: "('h, 'i, 'j) generat"   and generat' :: "('h, 'i, 'j) generat"   and f1 :: "'h \<Rightarrow> 'k"   and g1 :: "'h \<Rightarrow> 'k"   and f2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'k"   and g2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'k" assumes "generat = generat'"   and "\<And>x1. generat' = Pure x1 \<Longrightarrow> f1 x1 = g1 x1"   and "\<And>x21 x22. generat' = IO x21 x22 \<Longrightarrow> f2 x21 x22 = g2 x21 x22" shows "(case generat of Pure x1 \<Rightarrow> f1 x1 | IO x21 x22 \<Rightarrow> f2 x21 x22) = (case generat' of Pure x1 \<Rightarrow> g1 x1 | IO x21 x22 \<Rightarrow> g2 x21 x22)"
dperm_inv: fixes n :: "nat" shows "dist_perm (s_A n) (s_mtf n)"
pmf_eq_0_set_pmf: fixes M :: "'e pmf"   and p :: "'e" shows "(pmf M p = 0) = (p \<notin> set_pmf M)"
unitor_coincidence_1: fixes a :: "'a" assumes "B.obj a" shows "B.lunit a = \<i>[a]"
well_related_singleton_refl: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "less_eq x x" shows "well_related_set {x} less_eq"
IH_1: shows "setops\<^sub>s\<^sub>s\<^sub>t (map snd B) \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t (map snd (A @ B))"
measurable_CI: fixes n :: "nat" shows "\<mu>.I n \<in> borel_measurable (restrict_space borel {0<..<1})"
seq_comp_weakly_sequential: fixes P :: "'c option process"   and Q :: "'c option process" assumes "weakly_sequential P"   and "weakly_sequential Q" shows "weakly_sequential (P ; Q)"
w_nonneg: shows "AE x in M. 0 \<le> w x"
simps_3: fixes S :: "'node SDG_node set"   and cs :: "'edge list"   and a :: "'edge"   and as :: "'edge list" assumes "\<not> slice_edge S cs a" shows "slice_edges S cs (a # as) = slice_edges S (upd_cs cs [a]) as"
eq_rat_1: fixes b :: "int"   and d :: "int"   and a :: "int"   and c :: "int" assumes "b \<noteq> 0"   and "d \<noteq> 0" shows "(Rat.Fract a b = Rat.Fract c d) = (a * d = c * b)"
gcd_mult_ddvd: fixes d :: "'a"   and a :: "'a"   and b :: "'a" shows "d * gcd a b ddvd gcd (d * a) (d * b)"
disc_iff_1: fixes b :: "'b"   and xs :: "'a llist" shows "is_TNil (tllist_of_llist b xs) = lnull xs"
Collect_imp_eq: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
has_codomain_iff_arr: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "(codomains f \<noteq> {}) = arr f"
IH: fixes s :: "'s" shows "forward_ix_rec (IArray os) s t 0 = forward s t os"
simps_3: fixes ao :: "'a option"   and n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "proper_interval_set_Compl_aux ao n (x # xs) [] = (let m = CARD('a) - n; (len_x, x') = length_last (x # xs) in m \<noteq> len_x \<and> (m = len_x + 1 \<longrightarrow> \<not> proper_interval (Some x') None))"
cases: fixes a1 :: "'b com option \<times> 'b"   and a2 :: "'b com option \<times> 'b" assumes "a1 -c\<rightarrow> a2" obtains f :: "'b \<Rightarrow> 'b" and s :: "'b" where "a1 = (Some (Basic f), s__)" and "a2 = (None, f s__)"   | P0 :: "'b com" and s :: "'b" and t :: "'b" and P1 :: "'b com" where "a1 = (Some (Seq P0__ P1__), sa__)" and "a2 = (Some P1__, t__)" and "(Some P0__, sa__) -c\<rightarrow> (None, t__)"   | P0 :: "'b com" and s :: "'b" and P2 :: "'b com" and t :: "'b" and P1 :: "'b com" where "a1 = (Some (Seq P0 P1a__), sb__)" and "a2 = (Some (Seq P2__ P1a__), ta__)" and "(Some P0, sb__) -c\<rightarrow> (Some P2__, ta__)"   | s :: "'b" and b :: "'b set" and P1 :: "'b com" and P2 :: "'b com" where "a1 = (Some (Cond b__ P1b__ P2a__), sc__)" and "a2 = (Some P1b__, sc__)" and "sc__ \<in> b__"   | s :: "'b" and b :: "'b set" and P1 :: "'b com" and P2 :: "'b com" where "a1 = (Some (Cond ba__ P1 P2), sd__)" and "a2 = (Some P2, sd__)" and "sd__ \<notin> ba__"   | s :: "'b" and b :: "'b set" and P :: "'b com" where "a1 = (Some (While bb__ P__), se__)" and "a2 = (None, se__)" and "se__ \<notin> bb__"   | s :: "'b" and b :: "'b set" and P :: "'b com" where "a1 = (Some (While bc__ Pa__), sf__)" and "a2 = (Some (Seq Pa__ (While bc__ Pa__)), sf__)" and "sf__ \<in> bc__"   | s :: "'b" and b :: "'b set" and P :: "'b com" and t :: "'b" where "a1 = (Some (Await b P), s)" and "a2 = (None, t)" and "s \<in> b" and "(Some P, s) -c*\<rightarrow> (None, t)"
assms_2: shows "u \<rightarrow>\<^sub>\<beta>\<^sup>* v"
prj_chine_assoc_2: shows "cods.Prj\<^sub>1\<^sub>0 \<cdot> cods.chine_assoc = cods.Prj\<^sub>0\<^sub>1"
fl_dghm_surj_on_Hom: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "\<GG>\<lparr>ArrMap\<rparr> `\<^sub>\<circ> Hom \<BB> a b = Hom \<CC> (\<GG>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>) (\<GG>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>)"
count_roots_with_def: fixes P :: "'a \<Rightarrow> bool"   and p :: "'a poly" shows "count_roots_with P p = (\<Sum>x\<in>roots_with P p. order x p)"
finite_pred_constr_symvars: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat" assumes "RedBlack prb"   and "finite_RedBlack prb"   and "rv \<in> red_vertices prb" shows "\<forall>e\<in>pred (confs prb rv). finite (Bexp.vars e)"
assms_1: shows "f contour_integrable_on part_circlepath c r a b"
cong_iff_dvd_diff: 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 uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and m :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo a b m = dvd.dvd times m (minus a b)"
VDMConseq: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> c : A"   and "\<forall>s t. A s t \<longrightarrow> B s t" shows "G \<rhd> c : B"
lookup_update_neq: fixes k :: "'c"   and k' :: "'c"   and v :: "'d"   and m :: "('c, 'd) mapping" assumes "k \<noteq> k'" shows "Mapping.lookup (Mapping.update k v m) k' = Mapping.lookup m k'"
dictator_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and j :: "'c" shows "dictator scf A Is j \<equiv> j \<in> Is \<and> (\<forall>x\<in>A. \<forall>y\<in>A. decisive scf A Is {j} x y)"
inj_ab_group_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_ab_group_add_hom hom \<equiv> ab_group_add_hom hom \<and> inj_group_add_hom hom"
imageI: fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'c" assumes "x \<in> A" shows "f x \<in> f ` A"
that: fixes v :: "'a vec" assumes "v \<in> carrier_vec nc"   and "v \<noteq> 0\<^sub>v nc"   and "A *\<^sub>v v = 0\<^sub>v n" shows "thesis"
zero: shows "\<forall>v\<in>carrier_vec n. inner_prod v (A *\<^sub>v v) = 0"
is_unit_gauss_int_iff: fixes x :: "gauss_int" shows "is_unit x = (x \<in> {1, - 1, gauss_i, - gauss_i})"
span_base: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and S :: "'e set" assumes "module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
dAmf0: shows "A - f 0 \<in> carrier_mat d d"
free_tv_le_new_tv: fixes n :: "nat"   and t :: "'b"   and m :: "nat" assumes "new_tv n t"   and "m \<in> free_tv t" shows "m < n"
lmap_lconcat: fixes f :: "'b \<Rightarrow> 'a"   and xss :: "'b llist llist" shows "lmap f (lconcat xss) = lconcat (lmap (lmap f) xss)"
length_takeWhile_less: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. \<not> P x" shows "length (takeWhile P xs) < length xs"
abs_eq: shows "(\<cdot>) = (\<lambda>f g. Abs_nd_fun (Rep_nd_fun f \<circ>\<^sub>K Rep_nd_fun g))"
iso_class_is_nonempty: fixes F :: "'a set" assumes "B.is_iso_class F" shows "F \<noteq> {}"
1: fixes p' :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and q' :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "(p', q') \<in>\<^sub>p snd ` set ps"   and "fst p' \<noteq> 0"   and "fst q' \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"
ftimesI1: fixes x :: "V"   and a :: "V"   and b :: "V"   and A :: "V"   and B :: "V" assumes "x = [a, b]\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> B" shows "x \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B"
rinf\<^sub>2: shows "r \<in> \<V> f\<^sub>2"
AboveS_decr: fixes A :: "'a set"   and B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "A \<subseteq> B" shows "AboveS r B \<subseteq> AboveS r A"
eval_statement_no_jump: fixes Env :: "env"   and s0 :: "state"   and c :: "stmt"   and s1 :: "state"   and j :: "jump" assumes "prg Env\<turnstile>s0 \<midarrow>c\<rightarrow> s1"   and "jumpNestingOkS {} c"   and "fst s0 \<noteq> Some (Jump j)"   and "Env\<turnstile>c\<Colon>\<surd>"   and "wf_prog (prg Env)" shows "fst s1 \<noteq> Some (Jump j)"
rel_distinct_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y21 :: "'b"   and y22 :: "'b list" shows "\<not> list_all2 R (y21 # y22) []"
simps_16: fixes Vs :: "String.literal list"   and V :: "String.literal"   and T :: "ty"   and vo :: "'a val option"   and e :: "'a expr" shows "compE1 Vs {V:T=vo; e} = {length Vs:T=vo; compE1 (Vs @ [V]) e}"
assms_1: fixes x :: "'a"   and y :: "'b" assumes "phi x y" shows "chi x y"
dehomogenize_sum: fixes x :: "'b"   and f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and A :: "'d set" shows "dehomogenize x (sum f A) = (\<Sum>a\<in>A. dehomogenize x (f a))"
that: fixes y :: "'a" assumes "rep_nat_term y = (t, i)" shows "thesis"
tensor_preserves_ide: fixes a :: "'c"   and b :: "'c" assumes "C.ide a"   and "C.ide b" shows "C.ide (a \<otimes>\<^sub>C b)"
path_aux1a: fixes x :: "'a" assumes "forward_terminating_path x" shows "(x \<noteq> 0) = (end_points x \<noteq> 0)"
zero_min: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq zero x"
fmrestrict_fset_dom: fixes m :: "('a, 'b) fmap" shows "fmrestrict_fset (fmdom m) m = m"
Sup_insert: 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 a :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (a \<triangleright> A) = sup a (Sup A)"
intros_2: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and n :: "nat" shows "wf_term \<Sigma> (Bv n)"
Wbis_MtransT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>w d"   and "s \<approx> t"   and "(c, s) \<rightarrow>*t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>*t t'__" and "s' \<approx> t'__"   | d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t')" and "s' \<approx> t'" and "discr d'"
Fract_le_one_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(Rat.Fract a b \<le> 1) = (a \<le> b)"
transcendental_sin: fixes z :: "complex" assumes "algebraic z"   and "z \<noteq> 0" shows "\<not> algebraic (sin z)"
cond_D_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" assumes "hlde a b" shows "hlde.cond_D a b x y = (\<forall>l\<le>length b. take l b \<bullet> take l y \<le> a \<bullet> x)"
scalar_prod_left_unit: fixes v :: "'a vec"   and n :: "nat"   and i :: "nat" assumes "v \<in> carrier_vec n"   and "i < n" shows "unit_vec n i \<bullet> v = v $ i"
br'_inner_invar_imp_final: fixes q :: "'d"   and W :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<Sigma>' :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" assumes "q \<in> W"   and "br'_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) {} \<Sigma>'" shows "((Q, W, rcm), \<Sigma>') \<in> br'_step \<delta>"
S: shows "(s', s) \<in> \<langle>S\<rangle>set_rel"
suffix_nth: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "suffix k x n = x (k + n)"
top_on_post: fixes C :: "'a st option acom"   and X :: "char list set" assumes "top_on\<^sub>c C X" shows "top_on\<^sub>o (post C) X"
as_needed_asses_submap_exec_x: fixes v :: "'a"   and a :: "('a, 'b) fmap \<times> 'c"   and f :: "('a, 'b) fmap" assumes "v \<in> fmdom' (action_needed_asses a f)" shows "v \<in> fmdom' (fst a) \<and> v \<in> fmdom' f \<and> fmlookup (fst a) v = fmlookup f v"
old_iteratei_rule_insert_P: fixes m :: "'s"   and I :: "'u set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'u \<times> 'v \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "invar m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom (\<alpha> m) - it; \<alpha> m k = Some v; it \<subseteq> dom (\<alpha> m); I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom (\<alpha> m)) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom (\<alpha> m); it \<noteq> dom (\<alpha> m); \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iteratei m c f \<sigma>0)"
in_carrier: fixes n :: "nat" shows "set (replicate n \<zero>) \<subseteq> carrier R"
renaming: fixes xvec :: "name list"   and Tvec :: "'a list"   and p :: "(name \<times> name) list"   and T :: "'a" assumes "length xvec = length Tvec"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "(p \<bullet> xvec) \<sharp>* T" shows "T[xvec::=Tvec] = (p \<bullet> T)[(p \<bullet> xvec)::=Tvec]"
ss: shows "read_only \<S> \<subseteq> read_only \<S>'"
fmpred_alt_def: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap" shows "fmpred P m = fBall (fmdom m) (\<lambda>x. P x (the (fmlookup m x)))"
characteristic_weak_formula_is_characteristic': fixes Q :: "'state"   and P :: "'state" shows "Q \<Turnstile>\<^sub>S S_transform.characteristic_weak_formula P = S_transform.weakly_logically_equivalent P Q"
difference_supp: fixes ys :: "W.tvar list"   and xs :: "W.tvar list" shows "supp (xs - ys) = supp xs - supp ys"
per_alw: fixes r :: "'rule"   and steps :: "('state \<times> 'rule) stream" assumes "per r"   and "epath steps \<and> fst (shd steps) \<in> S" shows "alw (\<lambda>xs. holds (enabledAtStep r) xs \<longrightarrow> holds (\<lambda>step. snd step = r) xs \<or> nxt (holds (enabledAtStep r)) xs) steps"
sorted_wrt_dist_insort_key: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and q :: "(real, 'b) vec" assumes "sorted_wrt_dist p ps" shows "sorted_wrt_dist p (insort_key (\<lambda>q. dist q p) q ps)"
Int_stable: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "algebra \<Omega> M" shows "Int_stable M"
fresh_1: fixes a :: "name"   and x2 :: "'d"   and x1 :: "'d" shows "a \<sharp> x2\<lparr>x1\<rparr> = (a \<sharp> x2 \<and> a \<sharp> x1)"
l: shows "(f \<circ> r) \<longlonglongrightarrow> l"
simps_2: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and i :: "'d"   and x :: "'e"   and y :: "'e" shows "rel_comp_pow r [i] x y = r i x y"
spanning_subset_independent: fixes U\<^sub>M :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'d set" assumes "module_on U\<^sub>M scale"   and "A \<subseteq> U\<^sub>M"   and "B \<subseteq> A"   and "\<not> module_on.dependent scale A"   and "A \<subseteq> module_on.span scale B" shows "A = B"
g_def: shows "g \<equiv> \<lambda>x. 1 / ln (b * x) * ln (1 + hb / b * ln x powr (- 1 - e))"
chamber_image_complement_subset: fixes D :: "'a set" assumes "D \<in> folding.\<C> - f \<turnstile> folding.\<C>" shows "order.greater_eq {B \<in> folding.\<C>. order.greater (chamber_distance B (f ` D)) (chamber_distance B D)} (folding.\<C> - f \<turnstile> folding.\<C>)"
bop_const_2: fixes u :: "('e, 'f) uexpr"   and v :: "('g, 'f) uexpr" shows "bop (\<lambda>x y. x) u v = u"
a_comp_simp: fixes x :: "'a"   and y :: "'a" shows "(ad x \<oplus> ad y) \<cdot> d x = ad y \<cdot> d x"
infinite_dimension_hom: fixes K :: "'a list set"   and E :: "'a list set" assumes "subfield K (K [X])"   and "\<one>\<^bsub>Rupt A P\<^esub> \<noteq> \<zero>\<^bsub>Rupt A P\<^esub>"   and "inj_on (rupture_surj A P) E"   and "subalgebra K E (K [X])"   and "UP_K.infinite_dimension K E" shows "Rupt.infinite_dimension (rupture_surj A P ` K) (rupture_surj A P ` E)"
length_zip: fixes xs :: "'e list"   and ys :: "'f list" shows "length (zip xs ys) = min (length xs) (length ys)"