hcju/isabelleps huggingface dataset:
simps_8: fixes a :: "('a, 'b, 'c) hp" shows "SIGP a** = SIGP a"
FDERIV_n_floatarith_def: fixes fa :: "floatarith"   and xs :: "nat list"   and ds :: "floatarith list"   and n :: "nat" shows "FDERIV_n_floatarith fa xs ds n = ((\<lambda>x. FDERIV_floatarith x xs ds) ^^ n) fa"
HEndPhase1_HInv4a2: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4a2 s p"   and "Inv2a s" shows "HInv4a2 s' p"
in_n_or_max_gt: shows "D' \<in> N \<and> S D' = {#} \<or> Max (atms_of D') < Max (atms_of D)"
simps_7: fixes x3 :: "'e"   and x1 :: "'d" shows "Fault x3 \<noteq> Normal x1"
disjnt_insert2: fixes Y :: "'c set"   and a :: "'c"   and X :: "'c set" shows "disjnt Y (insert a X) = (a \<notin> Y \<and> disjnt Y X)"
iLastWeak_iAll_conv: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLastWeak t0 I P = (\<box> t {iprev t0 I} \<down>< t0. P t)"
map_four_block_mat: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and C :: "'b mat"   and nr2 :: "nat"   and D :: "'b mat"   and f :: "'b \<Rightarrow> 'c" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "map_mat f (four_block_mat A B C D) = four_block_mat (map_mat f A) (map_mat f B) (map_mat f C) (map_mat f D)"
le: fixes n :: "nat"   and x :: "'c" assumes "N \<le> n \<and> x \<in> A" shows "norm (f x n) \<le> g n"
maux_axioms: shows "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"
pullback_bottom: fixes x :: "'a" assumes "domain.has_bottom P"   and "x \<in> P"   and "f x = codbot" shows "x = dombot"
3: shows "\<forall>v\<in>T. \<forall>u\<in>subterms v. u \<notin> subterms (fresh_pgwt (\<Union> (funs_term ` T)) (\<Gamma> s))"
simps_2: fixes k :: "'c"   and ks :: "'c list"   and vo :: "'d option"   and ts :: "('c \<times> ('c, 'd) trie) list" shows "delete_trie (k # ks) (Trie vo ts) = (case map_of ts k of None \<Rightarrow> Trie vo ts | Some t \<Rightarrow> let t' = delete_trie ks t in if is_empty_trie t' then Trie vo (AList.delete_aux k ts) else Trie vo (AList.update k t' ts))"
assms: shows "b_graph.is_lasso (degeneralize_ext m) L"
assms_1: shows "P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P'"
conjE: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "dynamic_component.eval tCMP active cid t t' n (dynamic_component.conj \<gamma> \<gamma>')" shows "dynamic_component.eval tCMP active cid t t' n \<gamma> \<and> dynamic_component.eval tCMP active cid t t' n \<gamma>'"
linorder_cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" obtains "less x y"   | "x = y"   | "less y x"
converged_pd_cs_single: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and m :: "nat"   and m' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "l < m"   and "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m'"   and "\<pi> 0 = \<pi>' 0"   and "\<pi> m pd\<rightarrow> \<pi> l"   and "cs\<^bsup>\<pi>\<^esup> l = [\<pi> l]" shows "\<exists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
check_induct'': fixes s :: "('i, 'a) state"   and P :: "('i, 'a) state \<Rightarrow> bool" assumes "\<nabla> s"   and "\<triangle> (\<T> s)"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<U> s \<Longrightarrow> P s"   and "\<And>s. \<lbrakk>\<nabla> s; \<triangle> (\<T> s); \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s; \<diamond> s; \<not> \<U> s; min_lvar_not_in_bounds s = None\<rbrakk> \<Longrightarrow> P s"   and "\<And>s x\<^sub>i dir I. \<lbrakk>dir = Positive \<or> dir = Negative; \<nabla> s; \<triangle> (\<T> s); \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s; \<diamond> s; \<not> \<U> s; min_lvar_not_in_bounds s = Some x\<^sub>i; \<lhd>\<^sub>l\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x\<^sub>i) (Direction.LB dir s x\<^sub>i); min_rvar_incdec dir s x\<^sub>i = Inl I\<rbrakk> \<Longrightarrow> P (set_unsat I s)" shows "P (check s)"
pt2: fixes pi1 :: "('e \<times> 'e) list"   and pi2 :: "('e \<times> 'e) list"   and x :: "'d" assumes "pt TYPE('d) TYPE('e)" shows "(pi1 @ pi2) \<bullet> x = pi1 \<bullet> pi2 \<bullet> x"
dghm_is_arr_HomCod_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
span_add: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "module scale"   and "x \<in> module.span scale S"   and "y \<in> module.span scale S" shows "x + y \<in> module.span scale S"
f_mono: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "f m < f n"
ZeroHeightReachableElimAux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, r) \<in> Reachable"   and "0 = k" shows "r = s"
cases: fixes x :: "exp" obtains x :: "Vars.var" and e :: "exp" where "xa__ = Lam [xb__]. ea__"   | e :: "exp" and x :: "Vars.var" where "xa__ = App e xc__"   | x :: "Vars.var" where "xa__ = Var x"   | as :: "(Vars.var \<times> exp) list" and body :: "exp" where "xa__ = Terms.Let as body"   | b :: "bool" where "xa__ = Bool b"   | scrut :: "exp" and e1 :: "exp" and e2 :: "exp" where "xa__ = (scrut ? e1 : e2)"
sel_2: fixes x1 :: "symbol \<times> symbol list"   and x2 :: "nat"   and x3 :: "nat"   and x4 :: "nat" shows "item_dot (Item x1 x2 x3 x4) = x2"
vimage_eq_UN: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'c set" shows "f -` B = (\<Union>y\<in>B. f -` {y})"
weak_partial_order_subset: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_partial_order L"   and "A \<subseteq> carrier L" shows "weak_partial_order (L\<lparr>carrier := A\<rparr>)"
p2_5: fixes x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set" assumes "(x1, K1) \<in> p2"   and "(x2, K2) \<in> p2"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
assms: shows "Di \<in> set (subseqs (dbproj i D))"
Err_in_err: fixes A :: "'b set" shows "Err \<in> err A"
integrable_std_normal_moment: fixes k :: "nat" shows "integrable lborel (\<lambda>x. std_normal_density x * x ^ k)"
assms_4: fixes var :: "Vars.var"   and exp :: "exp" assumes "{atom var} \<sharp>* c"   and "y = Lam [var]. exp" shows "P"
assms_1: shows "f \<in> space M \<rightarrow> space N"
q_def: shows "q = (\<lambda>f. \<Sum> {h \<in> hom_components (q' f). poly_deg h + poly_deg f = poly_deg p} when f \<in> F')"
coincide_on_eq: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and e :: "'b equation" assumes "coincide_on \<sigma> \<eta> (vars_of_eq e)" shows "subst_equation e \<sigma> = subst_equation e \<eta>"
supp_Conj\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]" shows "supp (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>) = supp tset\<^sub>\<alpha>"
replacement_fun: fixes x :: "hf"   and f :: "hf \<Rightarrow> hf" shows "\<exists>z. \<forall>v. (v \<^bold>\<in> z) = (\<exists>u. u \<^bold>\<in> x \<and> v = f u)"
main': shows "(r = 0 \<or> degree r < degree g) \<and> smult (lead_coeff g ^ (1 + length (coeffs f) - length (coeffs g))) (g * 0 + f) = g * q + r"
transp_sequence: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and seq :: "nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat" assumes "transp r"   and "\<And>i. r (seq (Suc i)) (seq i)"   and "i < j" shows "r (seq j) (seq i)"
Int_emptyI: fixes A :: "'f set"   and B :: "'f set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
Dom_f'': shows "\<CC>\<lparr>Dom\<rparr>\<lparr>f''\<rparr> = a''"
finfun_const_apply: fixes b :: "'a"   and a :: "'b" shows "(K$ b) $ a = b"
cancel_sym: fixes a :: "bool \<times> 'e"   and b :: "bool \<times> 'e" assumes "canceling a b" shows "canceling b a"
int_of_uint64_shift: fixes n :: "uint64"   and k :: "nat" shows "int_of_uint64 (n >> k) = int_of_uint64 n div 2 ^ k"
cases: fixes a1 :: "(C, BD) sequent list"   and a2 :: "(C, BD) sequent" assumes "(a1, a2) \<in> g3mod1" obtains A :: "(C, BD) form" where "a1 = [ \<Empt> \<Rightarrow>* \<LM> Aa__  \<RM>]" and "a2 = ( \<Empt> \<Rightarrow>* \<LM> \<box> Aa__  \<RM>)"   | A :: "(C, BD) form" where "a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>]" and "a2 = ( \<LM> \<diamond> A  \<RM> \<Rightarrow>* \<Empt>)"
h: shows "steps0 (s, l, r) B (Suc stp) = (s', l', r')"
prod_assn_def: fixes P1 :: "'e \<Rightarrow> 'f \<Rightarrow> assn"   and P2 :: "'g \<Rightarrow> 'h \<Rightarrow> assn"   and a :: "'e \<times> 'g"   and c :: "'f \<times> 'h" shows "(P1 \<times>\<^sub>a P2) a c \<equiv> case (a, c) of ((a1, a2), c1, c2) \<Rightarrow> P1 a1 c1 * P2 a2 c2"
subset_diff: fixes B :: "'c set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "B \<subseteq> A"   and "finite A" shows "sum g A = sum g (A - B) + sum g B"
d_OUT_split_Vertex: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" shows "d_OUT (split f) (Vertex x) = d_OUT f x"
sat_preserving_redundancy_criterion_def: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" shows "sat_preserving_redundancy_criterion \<Gamma> Rf Ri \<equiv> sat_preserving_inference_system \<Gamma> \<and> redundancy_criterion \<Gamma> Rf Ri"
core_closure_diff_empty: fixes A :: "'a set" assumes "A \<subseteq> carrier" shows "core (closure A - A) = {}"
nofail_simps_3: fixes x :: "'a" shows "nofail (RETURN x) = True"
assms: fixes Q :: "ccs" shows "(Q, Q \<parallel> \<zero>) \<in> Rel"
preserves_reflects_arr: fixes f :: "'c \<times> 'c" shows "C.hseq (fst f) (snd f) = C.VV.arr f"
ex231a: fixes x :: "'a" shows "((1::'a) \<sqinter> x * x\<^sup>T) * x = x"
assms_2: shows "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
scaleR_eq_iff: fixes b :: "'c"   and u :: "real"   and a :: "'c" shows "(b + u *\<^sub>R a = a + u *\<^sub>R b) = (a = b \<or> u = 1)"
vars_add_Var_subset: fixes p' :: "'a mpoly"   and x :: "nat"   and \<V> :: "nat set" assumes "vars (p' + Var x) \<subseteq> \<V>" shows "vars p' \<subseteq> insert x \<V>"
order_iff_strict: fixes b :: "'b list"   and a :: "'b list" shows "??.Sublist.prefix b a = (strict_prefix b a \<or> a = b)"
ccSUP_least: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'a"   and u :: "'a" assumes "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> f i \<le> u" shows "Sup (f ` A) \<le> u"
compact_sup_maxdistance: fixes S :: "'b set" assumes "compact S"   and "S \<noteq> {}" shows "\<exists>x\<in>S. \<exists>y\<in>S. \<forall>u\<in>S. \<forall>v\<in>S. dist u v \<le> dist x y"
nth_default_coeffs_eq: fixes p :: "'a poly" shows "nth_default (0::'a) (coeffs p) = coeff p"
map_zip_map2: fixes f :: "'c \<times> 'd \<Rightarrow> 'b"   and xs :: "'c list"   and g :: "'e \<Rightarrow> 'd"   and ys :: "'e list" shows "map f (zip xs (map g ys)) = Map2.map2 (\<lambda>x y. f (x, g y)) xs ys"
eq_fold: fixes x :: "'b"   and A :: "'b fset" shows "fMax (finsert x A) = ffold max x A"
subst_apply_labeled_stateful_strand_def: fixes S :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list"   and \<theta> :: "'g \<Rightarrow> ('f, 'g) Term.term" shows "S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
bounded_antilinearI: fixes f :: "'d \<Rightarrow> 'e"   and K :: "real" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>C b) = cnj r *\<^sub>C f b"   and "\<forall>x. norm (f x) \<le> norm x * K" shows "bounded_antilinear f"
simps_1: fixes f :: "'c \<Rightarrow> 'c" shows "f ^^ 0 = id"
h4: shows "satisfy_equation_R p qs (get_subsets_R (reduce_system_R p (qs, matrix, subsets, signs))) (get_signs_R (reduce_system_R p (qs, matrix, subsets, signs)))"
l2_inv2D: fixes B :: "agent"   and A :: "agent"   and gny :: "msg"   and Ra :: "fid_t"   and x :: "l2_state" assumes "Auth B A \<langle>Number 0, gny, Exp Gen (NonceF (Ra $ nx))\<rangle> \<in> chan x"   and "A \<notin> bad x \<and> B \<notin> bad x"   and "x \<in> l2_inv2" shows "\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress x Rb) {xny, xgnx, xgny, xsk} \<and> gny = Exp Gen (NonceF (Rb $ ny)) \<and> guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx)))"
ex_countable_subbasis: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.second_countable_topology open" shows "\<exists>B. countable B \<and> open = generate_topology B"
idiff_self: fixes n :: "enat" assumes "n \<noteq> \<infinity>" shows "n - n = 0"
eq0: shows "inner_prod (psi_l l) (psi_l l) = inner_prod (alpha_l l \<cdot>\<^sub>v \<alpha>) (psi_l l) + inner_prod (beta_l l \<cdot>\<^sub>v \<beta>) (psi_l l)"
read_only_reads_unowned: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'p"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'p memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'p"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'p memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<O>\<^sub>i) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) = {}"
simps_1: fixes s :: "'a stream" shows "s !! 0 = shd s"
diff_intersect_right_idem: fixes M :: "'b multiset"   and N :: "'b multiset" shows "M - N \<inter># M = M - N"
IH_2: shows "\<forall>t1 t2. t1 \<cdot> t2 \<prec> s2 \<longrightarrow> t1 \<prec> s2 \<and> t2 \<prec> s2"
store_def: fixes m :: "mem"   and n :: "nat"   and off :: "nat"   and bs :: "bytes"   and l :: "nat" shows "store m n off bs l = (if n + off + l \<le> mem_size m then Some (write_bytes m (n + off) (bytes_takefill 0 l bs)) else None)"
simps_13: fixes prec :: "nat"   and x :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "float interval option list" shows "isDERIV_approx prec x (Powr a b) vs = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs \<and> (case approx prec a vs of None \<Rightarrow> False | Some ivl \<Rightarrow> 0 < lower ivl))"
nor_sim_1: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> (F \<^bold>\<or> G)" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> F"
append_vec_def: fixes v :: "'a vec"   and w :: "'a vec" shows "v @\<^sub>v w \<equiv> let n = dim_vec v; m = dim_vec w in vec (n + m) (\<lambda>i. if i < n then v $ i else w $ (i - n))"
filter_eq_iff: fixes F :: "'b filter"   and F' :: "'b filter" shows "(F = F') = (\<forall>P. eventually P F = eventually P F')"
division_ofD_4: fixes s :: "'c set set"   and i :: "'c set"   and K :: "'c set" assumes "s division_of i"   and "K \<in> s" shows "\<exists>a b. K = cbox a b"
\<psi>_6: fixes p :: "'a" assumes "p \<in> carrier" shows "\<psi> p \<in> atlas"
le_imp_neg_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b" shows "less_eq (uminus b) (uminus a)"
gkernTr11: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and a :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "a \<in> carrier F" shows "iim F G f {f a} = gker\<^bsub>F,G\<^esub> f \<bullet>\<^bsub>F\<^esub> a"
ZObis_Sym: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01 t" shows "t \<approx>01 s"
nsqn_invalidate_eq: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option"   and rsn :: "nat" assumes "dip \<in> kD rt"   and "dests dip = Some rsn" shows "nsqn (invalidate rt dests) dip = rsn - 1"
singleton_conv: fixes a :: "'e" shows "{x. x = a} = {a}"
empty_mult1: fixes a :: "'b"   and R :: "('b \<times> 'b) set" shows "({#}, {#a#}) \<in> mult1 R"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "countable (B x)"
distinct_1: fixes a1 :: "'d"   and a2 :: "'d"   and a1' :: "'d"   and list' :: "name list"   and a2' :: "'d" shows "a1\<lparr>a2\<rparr> \<noteq> a1'\<lparr>\<nu>*list'\<rparr>\<langle>a2'\<rangle>"
cone_decomp_exact_aux: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs"   and "cone_decomp T qs" shows "cone_decomp T (exact_aux k m qs)"
scale_one_on: fixes x :: "'a \<Rightarrow> real" assumes "x \<in> src.diff_fun_space" shows "1 *\<^sub>R x = x"
blinfun_bij2: fixes f :: "'a \<Rightarrow>\<^sub>L 'a"   and g :: "'a \<Rightarrow>\<^sub>L 'a" assumes "f o\<^sub>L g = id_blinfun" shows "bij (blinfun_apply g)"
set_bset_inject: fixes x :: "'a set['b]"   and y :: "'a set['b]" shows "(set_bset x = set_bset y) = (x = y)"
Un_Diff_cancel2: fixes B :: "'k set"   and A :: "'k set" shows "B - A \<union> A = B \<union> A"
hom_commute: fixes h :: "'b \<Rightarrow> 'b"   and N :: "'b set" assumes "\<And>x y. h (min x y) = min (h x) (h y)"   and "finite N"   and "N \<noteq> {}" shows "h (Min N) = Min (h ` N)"
labeled_entailment_lifting: fixes NL1 :: "('f \<times> 'l) set"   and NL2 :: "('f \<times> 'l) set" shows "(NL1 \<Turnstile>\<inter>\<G>L NL2) = (fst ` NL1 \<Turnstile>\<inter>\<G> fst ` NL2)"
path_hd_appendPath: fixes Cs :: "char list list"   and Cs' :: "char list list"   and 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 C :: "char list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P,C \<turnstile> Cs \<sqsubseteq> Cs' @\<^sub>p Cs"   and "last Cs' = hd Cs"   and "Cs \<noteq> []"   and "Cs' \<noteq> []"   and "wf_prog wf_md P" shows "Cs' = [hd Cs]"
converse_relcomp: fixes r :: "('i \<times> 'j) set"   and s :: "('j \<times> 'h) set" shows "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
simps: fixes a1 :: "('fun, 'var) Term.term set"   and a2 :: "('fun, 'var) Term.term" shows "(a1 \<turnstile> a2) = ((\<exists>t M. a1 = M \<and> a2 = t \<and> t \<in> M) \<or> (\<exists>T f M. a1 = M \<and> a2 = Fun f T \<and> length T = arity f \<and> public f \<and> (\<forall>x. x \<in> set T \<longrightarrow> M \<turnstile> x)) \<or> (\<exists>M t K T t\<^sub>i. a1 = M \<and> a2 = t\<^sub>i \<and> M \<turnstile> t \<and> Ana t = (K, T) \<and> (\<forall>x. x \<in> set K \<longrightarrow> M \<turnstile> x) \<and> t\<^sub>i \<in> set T))"
auto_ll_on_open_axioms: shows "auto_ll_on_open (- f) X"
conv_radius_cong': fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>\<^sub>F x in sequentially. f x = g x" shows "conv_radius f = conv_radius g"
c_redu_subst2': fixes M :: "trm"   and M' :: "trm"   and c :: "coname"   and y :: "name"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>c M'" shows "M{c:=y.P} \<longrightarrow>\<^sub>c M'{c:=y.P}"
neq_Nil_revE: fixes l :: "'b list" assumes "l \<noteq> []" obtains ll :: "'b list" and e :: "'b" where "l = ll @ [e]"
simps_1: fixes x :: "'b equation" shows "mk_lit pos x = Pos x"
inf1I: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a"   and B :: "'a \<Rightarrow> bool" assumes "A x"   and "B x" shows "(A \<sqinter> B) x"
multi_member_skip: fixes x :: "'b"   and XS :: "'b multiset"   and y :: "'b" assumes "x \<in># XS" shows "x \<in># {#y#} + XS"
assms: fixes x :: "'d" assumes "x \<in># A"   and "P x"   and "Q x" shows "False"
axis_index_def: fixes v :: "('b, 'a) vec" shows "axis_index v \<equiv> SOME i. v = axis i (1::'b)"
irreducible\<^sub>d_reflect_poly: fixes f :: "'a poly" assumes "coeff f 0 \<noteq> (0::'a)" shows "irreducible\<^sub>d (reflect_poly f) = irreducible\<^sub>d f"
shift_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (shift_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
rep_eq: fixes x :: "'a" shows "blinfun_apply (blinfun_mult_right x) = (*) x"
is_processT4: fixes P :: "'b process" shows "\<forall>s X Y. (s, Y) \<in> F P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> F P"
CallRedsThrowObj: fixes e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and ex :: "nat \<times> char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 Copt :: "char list option"   and M :: "char list"   and es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw ex,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw ex,s\<^sub>1\<rangle>"
simps: fixes i :: "nat" shows "initState (Friend i) = {Key (priEK (Friend i)), Key (priSK (Friend i)), Key (shrK (Friend i))} \<union> Key ` range pubK \<union> Key ` range pubSK"