hcju/isabelleps huggingface dataset:
prognosis_Let: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and cHeap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and e :: "exp"   and a :: "Arity"   and as :: "Arity list" assumes "CardinalityPrognosisLet prognosis cHeap Aheap"   and "atom ` domA \<Delta> \<sharp>* \<Gamma>"   and "atom ` domA \<Delta> \<sharp>* S"   and "edom ae \<subseteq> domA \<Gamma> \<union> upds S" shows "prognosis (Aheap \<Delta> e\<cdot>a \<squnion> ae) as a (\<Delta> @ \<Gamma>, e, S) \<sqsubseteq> cHeap \<Delta> e\<cdot>a \<squnion> prognosis ae as a (\<Gamma>, Terms.Let \<Delta> e, S)"
hull_same: fixes S :: "'b set \<Rightarrow> bool"   and s :: "'b set" assumes "S s" shows "S hull s = s"
rel_resourceD: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and res1 :: "('c, 'e) resource"   and res2 :: "('d, 'f) resource" assumes "rel_resource A B res1 res2" shows "rel_fun A (rel_spmf (rel_prod B (rel_resource A B))) (run_resource res1) (run_resource res2)"
assms_1: shows "a \<in> set (insert_spec xs (e, i))"
gamma2_vec_diffable: fixes x :: "real" assumes "x \<in> {0..1} - s" shows "(\<gamma>2 \<circ> \<phi> has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and attach_shadow_root :: "(_) element_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root type_wf get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs \<equiv> (l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs \<and> l_set_mode type_wf set_mode set_mode_locs) \<and> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr set_shadow_root set_mode attach_shadow_root get_tag_name get_shadow_root"
estep_Pr2: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Pr n f g, xs, ls) # fs, rv)"   and "ls \<noteq> []"   and "length ls \<noteq> Suc (hd xs)"   and "rv \<up>"   and "recfn (length xs) (Pr n f g)" shows "estep_Pr (encode_config c) = encode_config (step c)"
Min_less_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less (linorder.Min less_eq A) x = (\<exists>a\<in>A. less a x)"
last_butlast_app: fixes as :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list"   and bs :: "('d \<times> 'e) list" assumes "butlast as = xs @ butlast bs"   and "bs \<noteq> []"   and "as \<noteq> []"   and "fst (last as) = fst (last bs)"   and "snd (last as) = snd (last bs)" shows "as = xs @ bs"
exhaust: fixes y :: "'c dual" obtains x :: "'c" where "y = dual x"
composite_in_hom_2: shows "\<guillemotleft>(cod \<mu> \<star> cod \<nu>) \<star> cod \<pi> : (local.dom (cod \<mu>) \<star> local.dom (cod \<nu>)) \<star> local.dom (cod \<pi>) \<Rightarrow> (cod (cod \<mu>) \<star> cod (cod \<nu>)) \<star> cod (cod \<pi>)\<guillemotright>"
FVTsub2: assumes "vid1 \<in> ODE_vars I (OVar vid1)" shows "FVT ($f fid1 (\<lambda>i. if i = vid1 then Var vid1 else Const 0)) \<subseteq> semBV I (OVar vid1)"
NORMAL_def: fixes t :: "trm" shows "NORMAL t \<equiv> \<nexists>t'. t \<mapsto> t'"
rebase_ge_def: shows "rebase_ge TYPE('a) TYPE('b) \<equiv> CARD('a) \<le> CARD('b)"
v_def: shows "v = \<lparr>ext = Abs_real_int (0, 3), lan = Abs_nat_int {0}, own = d\<rparr>"
ennreal_mult_divide_eq: fixes b :: "ennreal"   and a :: "ennreal" assumes "b \<noteq> 0"   and "b \<noteq> top_class.top" shows "a * b / b = a"
char_le_iff_nat_of_char: fixes x :: "char"   and y :: "char" shows "(x \<le> y) = (of_char x \<le> of_char y)"
length: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list" assumes "execution trans sends start exec trace" shows "length exec - 1 = length trace"
ms_weak_def: shows "ms_weak = ms_strict\<^sup>="
lepoll_iff: fixes A :: "'b set"   and B :: "'c set" shows "(A \<lesssim> B) = (\<exists>g. A \<subseteq> g ` B)"
sublist_Cons_right: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "sublist xs (y # ys) = (prefix xs (y # ys) \<or> sublist xs ys)"
has_bochner_integral_diff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and g :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "has_bochner_integral M f x"   and "has_bochner_integral M g y" shows "has_bochner_integral M (\<lambda>x. f x - g x) (x - y)"
sees: shows "P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>body\<rfloor> in D"
verts_del_arc: fixes a :: "'b" shows "verts (del_arc a) = verts G"
variant_names_amount: fixes S :: "variable set"   and n :: "nat"   and s :: "variable" assumes "finite S" shows "length (fst (variant_variables n s S)) = n"
permutes_inv_inv: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "inv (inv p) = p"
simps: fixes x :: "'c"   and y :: "'d"   and t :: "('c \<times> 'd) tree" shows "update x y t = (if t = \<langle>\<rangle> then \<langle>\<langle>\<rangle>, (x, y), \<langle>\<rangle>\<rangle> else case Splay_Map.splay x t of \<langle>l, a, r\<rangle> \<Rightarrow> if x = fst a then \<langle>l, (x, y), r\<rangle> else if x < fst a then \<langle>l, (x, y), \<langle>\<langle>\<rangle>, a, r\<rangle>\<rangle> else \<langle>\<langle>l, a, \<langle>\<rangle>\<rangle>, (x, y), r\<rangle>)"
implies_galois: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and implies :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.heyting_semilattice inf less_eq less implies" shows "less_eq (inf x y) z = less_eq x (implies y z)"
prems_1: shows "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) (x # xs)"
sample_vars_def: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "int pmf \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and X :: "'c \<Rightarrow> int pmf"   and A :: "'c set"   and m :: "'b" shows "prob_exp_base.sample_vars return bind get put sample X A m = Finite_Set.fold (\<lambda>x m. bind (memoization_base.memo return bind get put (prob_exp_base.sample_var return sample X) x) (\<lambda>_. m)) m A"
delay_bisimulation_measure: fixes \<mu>1 :: "'s1 \<Rightarrow> 's1 \<Rightarrow> bool"   and \<mu>2 :: "'s2 \<Rightarrow> 's2 \<Rightarrow> bool" assumes "wfP \<mu>1"   and "wfP \<mu>2" shows "delay_bisimulation_measure trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 \<mu>1 \<mu>2"
has_derivative_continuous_on: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x within s)" shows "continuous_on s f"
a_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
homf: fixes i :: "'a" assumes "i \<in> I" shows "f i \<in> hom (G i) (H i)"
simps_6: fixes I :: "interp"   and \<alpha> :: "game" shows "game_sem I \<alpha>** = (\<lambda>X. \<Inter> {Z. X \<union> game_sem I \<alpha> Z \<subseteq> Z})"
induct: fixes x :: "Vars.var list"   and S :: "Vars.var set"   and G :: "CoCalls"   and P :: "Vars.var list \<Rightarrow> bool" assumes "x \<in> valid_lists S G"   and "P []"   and "\<And>xs x. \<lbrakk>set xs \<subseteq> ccNeighbors x G; xs \<in> valid_lists S G; P xs; x \<in> S\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P x"
rem_implicit_pres_ops_enabled_1: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and s :: "nat \<Rightarrow> nat option" assumes "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op"   and "ast_problem.enabled prob name s" shows "ast_problem.enabled (rem_implicit_pres_ops prob) name s"
cases: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> a1 \<Rightarrow>\<^sub>z a2" obtains "a1 = []" and "a2 = Undecided"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule m__ Accept # rs__" and "a2 = Decision FinalAllow" and "matches \<gamma> m__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule ma__ Drop # rsa__" and "a2 = Decision FinalDeny" and "matches \<gamma> ma__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule mb__ Reject # rsb__" and "a2 = Decision FinalDeny" and "matches \<gamma> mb__ p"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" where "a1 = Rule mc__ Log # rsc__" and "a2 = ta__" and "matches \<gamma> mc__ p" and "\<Gamma>,\<gamma>,p\<turnstile> rsc__ \<Rightarrow>\<^sub>z ta__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" where "a1 = Rule md__ Empty # rsd__" and "a2 = tb__" and "matches \<gamma> md__ p" and "\<Gamma>,\<gamma>,p\<turnstile> rsd__ \<Rightarrow>\<^sub>z tb__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" and a :: "action" where "a1 = Rule me__ a # rse__" and "a2 = tc__" and "\<not> matches \<gamma> me__ p" and "\<Gamma>,\<gamma>,p\<turnstile> rse__ \<Rightarrow>\<^sub>z tc__"   | m :: "'c match_expr" and chain :: "char list" and rs\<^sub>1 :: "'c rule list" and m' :: "'c match_expr" and rs\<^sub>2 :: "'c rule list" and rrs :: "'c rule list" and t :: "state" where "a1 = Rule mf__ (Call chain__) # rrs__" and "a2 = td__" and "matches \<gamma> mf__ p" and "\<Gamma> chain__ = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2)" and "matches \<gamma> m' p" and "\<Gamma>,\<gamma>,p\<turnstile> rs\<^sub>1 \<Rightarrow>\<^sub>z Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> rrs__ \<Rightarrow>\<^sub>z td__"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and X :: "final_decision" and rrs :: "'c rule list" where "a1 = Rule mg__ (Call chaina__) # rrsa__" and "a2 = Decision X" and "matches \<gamma> mg__ p" and "\<Gamma> chaina__ = Some rsf__" and "\<Gamma>,\<gamma>,p\<turnstile> rsf__ \<Rightarrow>\<^sub>z Decision X"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and rrs :: "'c rule list" and t :: "state" where "a1 = Rule m (Call chain) # rrs" and "a2 = t" and "matches \<gamma> m p" and "\<Gamma> chain = Some rs" and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t"
iadd_is_0: fixes m :: "enat"   and n :: "enat" shows "(m + n = 0) = (m = 0 \<and> n = 0)"
left_right_impl_times: fixes a :: "'a"   and b :: "'a" shows "a * (a r\<rightarrow> b) = (a l\<rightarrow> b) * a"
f: fixes b :: "real" assumes "a \<le> b" shows "set_integrable lborel {a..b} f"
eq_eqvt: fixes p :: "perm"   and x :: "'b"   and y :: "'b" shows "p \<bullet> (x = y) = (p \<bullet> x = p \<bullet> y)"
test_kat_21_opp: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q" shows "x \<cdot> q \<le> p \<cdot> x \<longrightarrow> p \<cdot> x \<cdot> q = x \<cdot> q"
g: fixes w :: "'b" shows "g w \<ge>p zero_poly"
is_RELABEL_def: fixes pr_operation :: "pr_operation" shows "is_RELABEL pr_operation = (pr_operation = RELABEL)"
mismatchNil: fixes a :: "name"   and P :: "pi" shows "[a\<noteq>a]P \<sim>\<^sub>e \<zero>"
m: shows "\<lfloor>real n * f x\<rfloor> = int m"
subset_Collect_conv: fixes S :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(S \<subseteq> Collect P) = (\<forall>x\<in>S. P x)"
holomorphic_factor_puncture: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "not_essential f z"   and "\<exists>\<^sub>F w in at z. f w \<noteq> 0" shows "\<exists>!x. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int x \<and> g w \<noteq> 0)"
hmem_succ: fixes l :: "hf"   and k :: "hf" assumes "l \<^bold>\<in> k" shows "l \<^bold>\<in> succ k"
norm_blinfun_bound: fixes b :: "real"   and f :: "'d \<Rightarrow>\<^sub>L 'e" assumes "0 \<le> b"   and "\<And>x. norm (blinfun_apply f x) \<le> b * norm x" shows "norm f \<le> b"
stationary_distribution_iterate': fixes K :: "'a \<Rightarrow> 'a pmf"   and N :: "'a pmf"   and y :: "'a"   and n :: "nat" assumes "MC_syntax.stationary_distribution K N" shows "measure_pmf.prob N {y} = measure_pmf.expectation N (\<lambda>x. MC_syntax.p K x y n)"
xt1_6: fixes y :: "'al"   and x :: "'al"   and z :: "'al" assumes "y \<in> U"   and "x \<in> U"   and "z \<in> U"   and "y \<le>\<^sub>o\<^sub>w x"   and "z \<le>\<^sub>o\<^sub>w y" shows "z \<le>\<^sub>o\<^sub>w x"
sorted_list_of_fset_simps_2: fixes S :: "'a fset" shows "fset_of_list (sorted_list_of_fset S) = S"
negligible_frontier_interval: fixes a :: "'c"   and b :: "'c" shows "negligible (cbox a b - box a b)"
log_one: fixes a :: "real" shows "log a 1 = 0"
TR_sub_carr: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR1 R M N P \<union> TR2 R M N P \<union> TR3 R M N P \<union> TR4 R M N P \<subseteq> carrier P"
EF_lem3b: fixes x :: "'b"   and f :: "'b set" assumes "x \<in> EX' (f \<union> EX' (EF f))" shows "x \<in> EF f"
distinct_31: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stmt.If x51 x52 x53"
less_length_cut_same: fixes w :: "'b list"   and y :: "'b"   and p :: "nat"   and a :: "'b" assumes "(w @- sconst y) !! p = a" shows "a = y \<or> p < length (cut_same y (w @- sconst y)) \<and> w ! p = a"
refl: fixes succs :: "'a \<Rightarrow> 'a list"   and g :: "'a" shows "RTranClp succs g g"
e_def: shows "e \<equiv> \<lambda>i. let b = if i < length Basis_list then the (map_of (zip [0..<length Basis_list] Basis_list) i) else (0::'a) in (k - (l + u) /\<^sub>R 2) \<bullet> b / (((u - l) /\<^sub>R 2) \<bullet> b)"
assms_2: fixes A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and xs :: "'varSort"   and s :: "'sort" assumes "A \<in> set (ALF (xs, s))" shows "wlsAbs (xs, s) A"
out_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A Out B C" shows "Col A B C"
const_poly_dvd_iff: fixes c :: "'a"   and p :: "'a poly" shows "([:c:] dvd p) = (\<forall>n. c dvd poly.coeff p n)"
well2: shows "Multiset.Ball (hmsetmset (local.encode (Suc e) (n div base))) local.well_base\<^sub>h"
quotient_empty: fixes r :: "('c \<times> 'c) set" shows "{} // r = {}"
IH: shows "ord_Zp (x [^] na__) = int na__ * ord_Zp x"
bit_not_iff: 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 uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (not a) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> \<not> bit a n)"
relative_to_def: fixes P :: "'d set \<Rightarrow> bool"   and S :: "'d set" shows "P relative_to S \<equiv> \<lambda>T. \<exists>U. P U \<and> S \<inter> U = T"
induct: fixes x :: "'b"   and F :: "'b program"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> reachable F"   and "\<And>s. s \<in> Init F \<Longrightarrow> P s"   and "\<And>act s s'. \<lbrakk>act \<in> Acts F; s \<in> reachable F; P s; (s, s') \<in> act\<rbrakk> \<Longrightarrow> P s'" shows "P x"
cases: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> external_actions E" obtains aa :: "nat" and ad :: "'b" and M :: "String.literal" and vs :: "'b val list" and v :: "'b val" where "a = aa" and "aa \<in> actions E" and "action_obs E aa = NormalAction (ExternalCall ad M vs v)"
bigMax: shows "wmax (wmax (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2)) (wmax (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2)) \<equiv>\<^sub>E max (max (rl1 * rl2) (ru1 * rl2)) (max (rl1 * ru2) (ru1 * ru2))"
CFGExit_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b" shows "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit \<equiv> CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main \<and> CFGExit_axioms sourcenode targetnode kind valid_edge Entry get_proc Main Exit"
stable_Int: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> stable A"   and "F \<in> stable A'" shows "F \<in> stable (A \<inter> A')"
cons: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and x :: "'a" assumes "appending xs ys zs" shows "appending (x # xs) ys (x # zs)"
uniform_measure_exponential: fixes l :: "real"   and t :: "real" assumes "0 < l"   and "0 \<le> t" shows "uniform_measure (exponential l) {t<..} = distr (exponential l) borel ((+) t)"
axioms_2: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "pr_ideal R I addition multiplication zero unit" shows "ideal I R addition multiplication zero unit"
four_block_mat_def: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat"   and D :: "'b mat" shows "four_block_mat A B C D = (let nra = dim_row A; nrd = dim_row D; nca = dim_col A; ncd = dim_col D in Matrix.mat (nra + nrd) (nca + ncd) (\<lambda>(i, j). if i < nra then if j < nca then A $$ (i, j) else B $$ (i, j - nca) else if j < nca then C $$ (i - nra, j) else D $$ (i - nra, j - nca)))"
simps_5: fixes v :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators" shows "removeShadowRules1_alternative_rev [v \<oplus> va] = [v \<oplus> va]"
weakStatImp_def: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Rel :: "('e \<times> ('d, 'e, 'f) psi \<times> ('d, 'e, 'f) psi) set"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "env.weakStatImp substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P Rel Q \<equiv> \<forall>\<Psi>'. \<exists>Q' Q''. (Q, Q') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<tau> \<prec> P')}\<^sup>* \<and> assertionAux.FrameStatImp SImp' (assertionAux.insertAssertion SCompose' (assertionAux.extractFrame SCompose' SBottom' P) \<Psi>) (assertionAux.insertAssertion SCompose' (assertionAux.extractFrame SCompose' SBottom' Q') \<Psi>) \<and> (Q', Q'') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' (SCompose' \<Psi> \<Psi>') P (\<tau> \<prec> P')}\<^sup>* \<and> (SCompose' \<Psi> \<Psi>', P, Q'') \<in> Rel"
diff_0_right: fixes a :: "'a" shows "a - (0::'a) = a"
apd1: fixes x :: "'a" shows "ad x \<cdot> x = (0::'a)"
normalize_def: fixes l :: "(bool \<times> 'c) list" shows "Cancelation.normalize l = (THE l'. cancels_to l l' \<and> canceled l')"
scast_1: shows "SCAST('c \<rightarrow> 'b) 1 = (if LENGTH('c) = 1 then - 1 else 1)"
axioms_1: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme" assumes "vectorspace K V" shows "module K V"
assms_4: shows "inv x \<otimes> x' \<in> H"
image_subsetI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "\<P> f A \<subseteq> B"
INF_comp: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and z :: "'a" shows "\<Sqinter>(f ` A) * z = (\<Sqinter>a\<in>A. f a * z)"