hcju/isabelleps huggingface dataset:
spec3_assoc: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point"   and p\<^sub>3 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "is_generic p\<^sub>1 p\<^sub>2"   and "is_tangent p\<^sub>2 p\<^sub>3"   and "is_generic (local.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"   and "is_tangent p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3)" shows "local.add a p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3) = local.add a (local.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
fresh_star_set: fixes xs :: "'a set"   and c :: "'b" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>xs. x \<sharp> c"
indiff_imp_same_utility_value: fixes \<alpha> :: "real"   and \<beta> :: "real" assumes "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>"   and "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "mix_pmf \<beta> local.some_best local.some_worst \<approx>[\<R>] mix_pmf \<alpha> local.some_best local.some_worst" shows "\<beta> = \<alpha>"
bernoulli_stream_equiv: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 < p"   and "p < 1"   and "0 < q"   and "q < 1" shows "filt_equiv nat_filtration M N"
induct: fixes x :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool" assumes "x \<in> SMP M"   and "\<And>t. t \<in> M \<Longrightarrow> P t"   and "\<And>t t'. \<lbrakk>t \<in> SMP M; P t; t' \<sqsubseteq> t\<rbrakk> \<Longrightarrow> P t'"   and "\<And>t \<delta>. \<lbrakk>t \<in> SMP M; P t; wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>; wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)\<rbrakk> \<Longrightarrow> P (t \<cdot> \<delta>)"   and "\<And>t K T k. \<lbrakk>t \<in> SMP M; P t; Ana t = (K, T); k \<in> set K\<rbrakk> \<Longrightarrow> P k" shows "P x"
characteristic_weak_formula_card_aux: fixes P :: "'state" shows "|{distinguishing_weak_formula P Q |Q. \<not> P \<equiv>\<cdot> Q}| <o natLeq +c |UNIV|"
Int_left_commute: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
cClose1: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "!P \<longmapsto> a<\<nu>x> \<prec> P''"   and "x \<sharp> P"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a<\<nu>x> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>x>(P' \<parallel> P''))"
least: shows "\<forall>ja<ia. \<not> vec_to_iarray A !! ([mod_type_class.to_nat i..<IArray.length (vec_to_iarray A)] ! ja) \<noteq> (0::'a)"
B_wf: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>))"
CastThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and C :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
brw_cond_abs: fixes \<Sigma> :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and Qi :: "'d set" shows "(\<Sigma> \<in> brw_cond Qi) = (brw_\<alpha> \<Sigma> \<in> bre'_cond Qi)"
lookup_def: fixes t :: "('d \<times> 'c) tree"   and x :: "'d" shows "lookup t x = (case Splay_Map.splay x t of \<langle>\<rangle> \<Rightarrow> None | \<langle>xa, (a, b), xb\<rangle> \<Rightarrow> if x = a then Some b else None)"
phantom_rel_def: fixes A :: "('c \<times> 'd) set" shows "\<langle>A\<rangle>phantom_rel = {(Some x, y) |x y. (x, y) \<in> A} \<union> {uu_. \<exists>x y. uu_ = (None, y) \<and> (x, y) \<in> A}"
setLxy: shows "set (Lxy init {x, y}) = {x, y}"
trans: fixes b :: "'b Word.word"   and a :: "'b Word.word"   and c :: "'b Word.word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
W_def: fixes k :: "nat" shows "W k = wlp_while_n M0 M1 WS k P"
ide_dom: fixes f :: "'c" assumes "Cop.arr f" shows "Cop.ide (Cop.dom f)"
C: assumes "\<not> P \<sigma>"   and "\<sigma> \<Turnstile> B\<^sub>2 ;- M" shows "Q"
fes_8: shows "\<forall>fe'\<in>set ft. ofe_prio fe2 < ofe_prio fe' \<longrightarrow> \<not> \<gamma> (ofe_fields fe') p"
Diff_insert: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A\<setminus>insert a B = A\<setminus>B\<setminus>{a}"
nhop_update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "ip \<noteq> dip" shows "nhop (update rt dip (dsn, dsk, flag, hops, nhip)) ip = nhop rt ip"
known_ptr_not_element_ptr: fixes ptr :: "(_) object_ptr" assumes "\<not> is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"   and "ElementClass.known_ptr ptr" shows "NodeClass.known_ptr ptr"
map_indets_minus: fixes f :: "'e \<Rightarrow> 'c"   and r :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and s :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "map_indets f (r - s) = map_indets f r - map_indets f s"
peval_closed_1: fixes xs :: "'a list"   and e :: "pexpr" assumes "in_carrier xs" shows "peval xs e \<in> carrier R"
equivalence_mapI: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.quasi_inverses f g" shows "Span.equivalence_map f"
Quotient_msg: shows "Quotient (\<sim>) abs_msg rep_msg cr_msg"
elem_in_nhd: fixes U :: "'b set"   and T :: "'b set set"   and x :: "'b" assumes "U \<in> nhds\<^bsub>T\<^esub> x" shows "x \<in> U"
sel_3: fixes x3 :: "(Term_Class.term \<times> sterm) list" shows "clauses (Sabs x3) = x3"
Quotient_rep_equiv2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "Quotient R Abs Rep T"   and "T a b" shows "R (Rep b) a"
f_closed: fixes a :: "'ar" shows "f (Rep_a a) \<in> \<UU>B"
med_a0im1_ri_def: fixes o1 :: "m1_state" shows "med_a0im1_ri o1 \<equiv> \<lparr>signals = ri_runs2sigs (runs o1), corrupted = Domain (m1_state.leak o1) \<times> UNIV\<rparr>"
bound: fixes C :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and B :: "nat"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" assumes "bounded_gen_check C B"   and "B < x" shows "C (x # xs) s = False"
oreachable_local: fixes s :: "'c \<times> 'd"   and A :: "('c \<times> 'd, 'e) automaton"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and U :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'e"   and s' :: "'c \<times> 'd" assumes "s \<in> oreachable A S U"   and "(s, a, s') \<in> automaton.trans A"   and "S (fst s) (fst s') a" shows "s' \<in> oreachable A S U"
Can_Nmlize_Can: fixes t :: "'c local.term" assumes "Can t" shows "Can \<^bold>\<lfloor>t\<^bold>\<rfloor>"
assoc_scalar_prod_mult_mat_vec: fixes y :: "'a vec"   and n :: "nat"   and x :: "'a vec"   and m :: "nat"   and A :: "'a mat" assumes "y \<in> carrier_vec n"   and "x \<in> carrier_vec m"   and "A \<in> carrier_mat n m" shows "(A *\<^sub>v x) \<bullet> y = (A\<^sup>T *\<^sub>v y) \<bullet> x"
in_verts_G: shows "{u, v} \<subseteq> verts G"
simps_3: fixes x :: "name"   and a :: "name"   and P :: "pi" assumes "x \<sharp> a" shows "valid (a<x>.P) = valid P"
IF1rel_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "IF1rel R = (BNF_Def.Grp (IF1in {(x, y). R x y}) (IF1map fst))\<inverse>\<inverse> OO BNF_Def.Grp (IF1in {(x, y). R x y}) (IF1map snd)"
e1: shows "fst (trunc_bound_eucl p (alpha * fst X)) = alpha * fst X + e1"
codomain_unique: fixes b :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr"   and b' :: "('obj, 'obj) concrete_category.arr" assumes "b \<in> codomains f"   and "b' \<in> codomains f" shows "b = b'"
cases: fixes a1 :: "cld"   and a2 :: "char list" assumes "class_name a1 a2" obtains dcl :: "char list" and cl :: "cl" and fds :: "fd list" and meth_defs :: "meth_def list" where "a1 = cld_def dcl cl fds meth_defs" and "a2 = dcl"
vaflow1: fixes F :: "real \<Rightarrow> variable \<Rightarrow> real"   and \<theta> :: "trm"   and \<zeta> :: "real" assumes "solves_ODE I F x (the (usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta>))" shows "Uvariation (F \<zeta>) (F 0) {RVar x, DVar x}"
sgn_minus: fixes x :: "'a" shows "sgn (- x) = - sgn x"
assms_3: shows "\<not> stateful_strand_step.is_Send x"
resolution_deriv_def: shows "resolution_deriv = resolution_step\<^sup>*\<^sup>*"
SUP_bot_conv_2: fixes B :: "'e \<Rightarrow> 'c \<times> 'd"   and A :: "'e set" shows "(bot_prod = Sup_prod (B ` A)) = (\<forall>x\<in>A. B x = bot_prod)"
distinct_899: fixes x251 :: "t"   and x252 :: "unop_f"   and x311 :: "t"   and x312 :: "cvtop"   and x313 :: "t"   and x314 :: "sx option" shows "Unop_f x251 x252 \<noteq> Cvtop x311 x312 x313 x314"
entails_pure': fixes b :: "bool"   and Q :: "assn" shows "(\<not> (\<up> b \<Longrightarrow>\<^sub>A Q)) = (\<not> (emp \<Longrightarrow>\<^sub>A Q) \<and> b)"
cos_sin_eq: fixes x :: "'a" shows "cos x = sin (of_real pi / (2::'a) - x)"
degree_aform_err_to_aform_le: fixes X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" shows "degree_aform (aform_err_to_aform X n) \<le> max (degree_aform_err X) (Suc n)"
Abs_Tangle_Diagram_inverse: fixes y :: "wall" assumes "y \<in> {x. is_tangle_diagram x}" shows "Rep_Tangle_Diagram (Abs_Tangle_Diagram y) = y"
ccProd_below_cc_restr: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and S'' :: "Vars.var set"   and G :: "CoCalls" shows "(S G\<times> S' \<sqsubseteq> G G|` S'') = (S G\<times> S' \<sqsubseteq> G \<and> (S = {} \<or> S' = {} \<or> S \<subseteq> S'' \<and> S' \<subseteq> S''))"
cases: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and a :: "('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "intruder_model.public_ground_wf_term arity public a" obtains f :: "'a" and T :: "('a, 'b) Term.term list" where "a = Fun f T" and "public f" and "arity f = length T" and "\<And>t. t \<in> set T \<Longrightarrow> intruder_model.public_ground_wf_term arity public t"
and_foldl_sem_conv: fixes \<Gamma> :: "('a, 'b, 'c) formula list"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and I :: "('a, 'b, 'c) interp" assumes "\<And>\<phi>. List.member \<Gamma> \<phi> \<Longrightarrow> \<nu> \<in> fml_sem I \<phi>" shows "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma> TT)"
assms_2: shows "theta \<subseteq> WretrT (theta \<union> WbisT)"
dotprod_0_right: fixes xs :: "nat list" shows "xs \<bullet> zeroes (length xs) = 0"
term_distinct_4: fixes name :: "name"   and u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" shows "free name \<noteq> app u\<^sub>1 u\<^sub>2"
eval_a: shows "eval (Cn 1 r_rev_aux [Cn 1 r_dec [r_length], recf.Id 1 0]) [e] = eval r_rev_aux [e_length e - 1, e]"
elims: fixes x :: "nterm"   and y :: "(nterm \<times> nterm) option" assumes "unapp x = y" obtains t :: "nterm" and u :: "nterm" where "x = t $\<^sub>n u" and "y = Some (t, u)"   | v :: "name" where "x = Nconst v__" and "y = None"   | v :: "name" where "x = Nvar va__" and "y = None"   | v :: "name" and va :: "nterm" where "x = (\<Lambda>\<^sub>n v. va)" and "y = None"
map_comp: fixes g :: "'f \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'f"   and v :: "'g list" shows "map g (map f v) = map (g \<circ> f) v"
assms_1: shows "v \<in> fmdom' (action_needed_asses a s1)"
invP: shows "A \<TTurnstile> (PI \<rightarrow>) P"
lookup_atomize_poly: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'c \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'd"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.atomize_poly pair_of_term p) v = lookup (lookup p (term_powerprod.component_of_term pair_of_term v)) (term_powerprod.pp_of_term pair_of_term v)"
blinfun_componentwise: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'c" shows "f = (\<lambda>x. \<Sum>i\<in>Basis. blinfun_scaleR (blinfun_inner_left i) (blinfun_apply (f x) i))"
split_invar: fixes t :: "('c \<times> nat) tree" assumes "invar t" shows "split t = t"
simps_2: fixes x :: "char list" shows "Malformed!x = None"
andfold_MatchExp_not_disc_negatedI: fixes ms :: "'b match_expr list"   and disc :: "'b \<Rightarrow> bool"   and neg :: "bool" assumes "\<forall>m\<in>set ms. \<not> has_disc_negated disc neg m" shows "\<not> has_disc_negated disc neg (andfold_MatchExp ms)"
list_augment_def: fixes xs :: "'b list"   and k :: "nat"   and v :: "'b" shows "list_augment xs k v = (list_pad_out xs k)[k := v]"
llast_ldropn: fixes n :: "nat"   and xs :: "'c llist" assumes "enat n < llength xs" shows "llast (ldropn n xs) = llast xs"
map_last: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
cmp_eq: shows "component_of_term (lt g1) = component_of_term (lt g2)"
le_listI: fixes a :: "'a list"   and b :: "'a list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "length a = length b"   and "\<And>n. n < length a \<Longrightarrow> a ! n \<sqsubseteq>\<^bsub>r\<^esub> b ! n" shows "a [\<sqsubseteq>\<^bsub>r\<^esub>] b"
conc_fun_def: fixes R :: "('d \<times> 'e) set"   and m :: "'e nres" shows "\<Down> R m \<equiv> case m of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> RES (R\<inverse> `` X)"
lt1: shows "lt (monom_mult (lookup p (t \<oplus> lt q) / lc q) t q) = t \<oplus> lt q"
unique_idxD2: fixes bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h"   and f :: "('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "unique_idx bs data"   and "f \<in> set bs" shows "fst (snd f) < fst data"
effective_scalar_product_distributivity: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and v1 :: "'b list"   and v2 :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "plus_mult id f zer g inver"   and "length v1 = length v2"   and "length w1 = length w2" shows "f (plus_mult.scalar_product f zer g v1 v2) (plus_mult.scalar_product f zer g w1 w2) = plus_mult.scalar_product f zer g (mult.vec_vec_Tensor f v1 w1) (mult.vec_vec_Tensor f v2 w2)"
4: shows "\<forall>S\<in>{dual\<^sub>s\<^sub>t \<A>}. list_all tfr\<^sub>s\<^sub>t\<^sub>p S"
induct: fixes P :: "('b \<Rightarrow> 'b ttree) \<Rightarrow> 'b set \<Rightarrow> 'b ttree \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'b ttree"   and a1 :: "'b set"   and a2 :: "'b ttree"   and a3 :: "'b list" assumes "\<And>f T t. P f T t []"   and "\<And>f T t x xs. P (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x) xs \<Longrightarrow> P f T t (x # xs)" shows "P a0 a1 a2 a3"
free_vars_dist_dens_cexpr: fixes dst :: "pdf_dist"   and e1 :: "cexpr"   and e2 :: "cexpr" shows "free_vars (dist_dens_cexpr dst e1 e2) \<subseteq> free_vars e1 \<union> free_vars e2"
size_greater_than_select: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "size (filter_mset ((<) (select n xs)) (mset xs)) < length xs - n"
that: fixes \<T>' :: "'a set set" assumes "\<T>' \<subseteq> \<T>"   and "finite \<T>'"   and "S \<subseteq> \<Union> \<T>'" shows "thesis"
assms_7: shows "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"
hoare_diagram3: fixes D :: "'b \<times> 'b \<Rightarrow> 'e \<Rightarrow> 'e"   and X :: "'a \<Rightarrow> 'b \<Rightarrow> 'e"   and P :: "'b \<Rightarrow> 'e"   and Q :: "'b \<Rightarrow> 'e" assumes "dmono D"   and "\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (pair u i) j"   and "P \<sqsubseteq> \<Squnion>range X"   and "\<Squnion>range X \<sqinter> - grd (step D) \<sqsubseteq> Q" shows "\<Turnstile> P{| pt D |}Q"
hn_monadic_nfoldli: fixes P :: "assn"   and \<Gamma> :: "assn"   and Rl :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and l' :: "'c list"   and l :: "'d list"   and Rs :: "'e \<Rightarrow> 'f \<Rightarrow> assn"   and s' :: "'e"   and s :: "'f"   and c :: "'f \<Rightarrow> bool Heap"   and c' :: "'e \<Rightarrow> bool nres"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'f Heap"   and f' :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres" assumes "P \<Longrightarrow>\<^sub>t \<Gamma> * hn_ctxt (list_assn Rl) l' l * hn_ctxt Rs s' s"   and "\<And>s s'. hn_refine (\<Gamma> * hn_ctxt Rs s' s) (c s) (\<Gamma> * hn_ctxt Rs s' s) bool_assn (c' $ s')"   and "\<And>x x' s s'. hn_refine (\<Gamma> * hn_ctxt Rl x' x * hn_ctxt Rs s' s) (f x s) (\<Gamma> * hn_invalid Rl x' x * hn_invalid Rs s' s) Rs (f' $ x' $ s')" shows "hn_refine P (imp_nfoldli l c f s) (\<Gamma> * hn_invalid (list_assn Rl) l' l * hn_invalid Rs s' s) Rs (monadic_nfoldli $ l' $ c' $ f' $ s')"
image_zmset_pre: fixes t :: "'a"   and f :: "'b \<Rightarrow> 'a"   and M :: "'b zmultiset" assumes "t \<in>#\<^sub>z Auxiliary.image_zmset f M" shows "\<exists>m. m \<in>#\<^sub>z M \<and> f m = t"
transfer_operations_rl_6: fixes ni :: "nat"   and n :: "nat"   and xi :: "'b list"   and x :: "real list set"   and d :: "nat" assumes "(ni, n) \<in> nat_rel"   and "(xi, x) \<in> appr_rell"   and "length xi = d" shows "(RETURN (split_appr ops ni xi), split_spec_params d n x) \<in> \<langle>appr_rell \<times>\<^sub>r appr_rell\<rangle>nres_rel"
natToVertexList_nth_0: fixes es :: "nat list"   and f :: "face"   and v :: "nat" assumes "incrIndexList es |es| |vertices f|"   and "0 < |es|" shows "natToVertexList v f es ! 0 = Some f\<^bsup>es ! 0\<^esup> \<bullet> v"
simps_2: fixes n :: "nat"   and p :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "repeat (Suc n) p = p ;; repeat n p"
fps_is_right_unit_iff_zeroth_is_right_unit: fixes f :: "'a fps" shows "(\<exists>g. 1 = g * f) = (\<exists>k. (1::'a) = k * fps_nth f 0)"
is_red_monomialI: fixes c :: "'e"   and f :: "'t \<Rightarrow>\<^sub>0 'e"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and v :: "'t" assumes "c \<noteq> (0::'e)"   and "f \<in> F"   and "f \<noteq> 0"   and "lt f adds\<^sub>t v" shows "is_red F (monomial c v)"
simps: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and uu :: "nat"   and r :: "('b \<times> nat) tree" shows "del_max \<langle>l, (a, uu), r\<rangle> = (if r = \<langle>\<rangle> then (a, l) else let (x, r') = del_max r in (x, balanceR l a r'))"
not_red_map_in_map_not_red: fixes N :: "'f set" shows "\<G>_Fset N - Red_F_G (\<G>_Fset N) \<subseteq> \<G>_Fset (N - local.Red_F_\<G> N)"
scons_3: shows "sfilter\<cdot>p\<cdot>(sfilter\<cdot>q\<cdot>xs) = sfilter\<cdot>(\<Lambda> x. q\<cdot>x andalso p\<cdot>x)\<cdot>xs"
intros_3: fixes c :: "nat"   and u :: "'b \<Rightarrow> real"   and x :: "'b" assumes "real c < u x" shows "intv_elem x u (Greater c)"
scylinder_streams: fixes S :: "'b set"   and xs :: "'b set list" shows "scylinder S xs \<subseteq> streams S"
update_simps_10: fixes up :: "(nat, 'b \<times> 'a) mapping \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and s :: "('b, 'a) state" shows "\<V> (\<B>\<^sub>i\<^sub>l_update up s) = \<V> s"
swap_def: fixes a :: "atom"   and b :: "atom" shows "(a \<rightleftharpoons> b) = Abs_perm (if sort_of a = sort_of b then \<lambda>c. if a = c then b else if b = c then a else c else id)"
bisim_compose_def: fixes bisim1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and bisim2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s3 :: "'i" shows "(bisim1 \<circ>\<^sub>B bisim2) s1 s3 \<equiv> \<exists>s2. bisim1 s1 s2 \<and> bisim2 s2 s3"
sprod_r_distr: fixes a :: "'b"   and m :: "'a"   and n :: "'a" assumes "a \<in> carrier R"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "a \<cdot>\<^sub>s (m \<plusminus> n) = a \<cdot>\<^sub>s m \<plusminus> a \<cdot>\<^sub>s n"
sets_restrict_restrict_space: fixes M :: "'b measure"   and A :: "'b set"   and B :: "'b set" shows "sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A \<inter> B))"
dist_triangle_less_add: fixes x1 :: "'a"   and y :: "'a"   and e1 :: "real"   and x2 :: "'a"   and e2 :: "real" assumes "dist x1 y < e1"   and "dist x2 y < e2" shows "dist x1 x2 < e1 + e2"
hyps: fixes x :: "real" shows "cos (x + real n * (2 * pi)) = cos x"
steps_z_norm_sound': fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D"   and "[D']\<^bsub>v,n\<^esub> \<noteq> {}" shows "\<exists>Z. A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>* \<langle>l', Z\<rangle> \<and> Z \<noteq> {}"
nxtActI: fixes n :: "nat"   and c :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "n \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<and> \<parallel>c\<parallel>\<^bsub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub> \<and> \<not> (\<exists>k\<ge>n. latestAct_cond c t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> k)"
eFresh_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and ys :: "'varSort"   and y :: "'var"   and eX :: "'a withERR" assumes "\<not> eFresh MOD ys y eX" shows "\<exists>s X. eX = OK X \<and> \<not> igFresh MOD ys y X \<and> igWls MOD s X"
no_loose_bvar_imp_no_subst_bvs1: fixes t :: "Core.term"   and lev :: "nat"   and us :: "Core.term list" assumes "\<not> loose_bvar t lev" shows "subst_bvs1 t lev us = t"
matchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<sim> P"
linear_on_def: fixes S1 :: "'c set"   and S2 :: "'d set"   and s1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" shows "linear_on S1 S2 s1 s2 f \<equiv> module_hom_on S1 S2 s1 s2 f"
DomFunctor: fixes f :: "'c" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "dom\<^bsub>CatCod F\<^esub> (F ## f) = F @@ dom\<^bsub>CatDom F\<^esub> f"
not_rt_fresh_asI1: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "\<not> rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2" shows "\<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
plus_adds_pp_left: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s + t adds\<^sub>p v" shows "s adds\<^sub>p v"
wls_fresh_vsubst: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "wls s X" shows "fresh zs z (X #[y1 // y]_ys) = ((zs = ys \<and> z = y \<or> fresh zs z X) \<and> (fresh ys y X \<or> zs \<noteq> ys \<or> z \<noteq> y1))"
ord_elems: fixes a :: "'a" assumes "finite (carrier G)"   and "a \<in> carrier G" shows "{a [^] x |x. x \<in> UNIV} = {a [^] x |x. x \<in> {0..ord a - 1}}"