hcju/isabelleps huggingface dataset:
trimmed_uminus_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "trimmed (uminus F) = trimmed F"
rtrancl_fun_conv: fixes s :: "'a"   and t :: "'a"   and R :: "('a \<times> 'a) set" shows "((s, t) \<in> R\<^sup>*) = (\<exists>f n. f 0 = s \<and> f n = t \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> R))"
read_icache_mod_state: fixes s :: "'b sparc_state"   and addr :: "word32"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "read_instr_cache s addr = read_instr_cache (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr"
R_bound: fixes M :: "nat" assumes "N \<le> M" shows "\<bar>R M\<bar> \<le> 1 / 4"
elims_2: fixes x :: "\<I>"   and xa :: "nat"   and xb :: "nat \<times> 'a \<times> 'b" assumes "check_before x xa xb" obtains I :: "\<I>" and dt :: "nat" and t :: "nat" and a :: "'a" and b :: "'b" where "x = I" and "xa = dt" and "xb = (t, a, b)" and "enat t + right I < enat dt"
diag_mult: fixes x :: "'b"   and A :: "('b, 'c) sq_matrix" shows "diag x * A = x *\<^sub>S A"
point_indices_alt_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and t :: "int" assumes "incidence_system point_set block_collection" shows "incidence_system.point_indices point_set block_collection t = {int (block_collection index ps) |ps. int (card ps) = t \<and> ps \<subseteq> point_set}"
simE2_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto>\<^sup>^<Rel> Q"   and "eqvt Rel"   and "(P, Q) \<in> Rel"   and "Q \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
conv_def: fixes cs :: "instr list"   and s :: "item list"   and io :: "int option" shows "conv cs s io = (case io of None \<Rightarrow> unwind cs s | Some i \<Rightarrow> exec cs (VAL i # s))"
dgrad_p_set_leI: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "\<And>f. f \<in> F \<Longrightarrow> dgrad_p_set_le d {f} G" shows "dgrad_p_set_le d F G"
elims_3: fixes x :: "nat"   and xa :: "nat \<times> nat list \<times> 'c list" assumes "\<not> add_inv x xa" obtains n :: "nat" and u :: "nat" and ns :: "nat list" and xs :: "'c list" where "x = n" and "xa = (u, ns, xs)" and "\<not> (foldl (+) 0 ns = n \<and> length xs = n)"
monad_state_prob_optionT: fixes return :: "'b option \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b option \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and sample :: "'e pmf \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_state_prob return bind get put sample" shows "monad_state_prob (return_option return) (bind_option return bind) (get_option get) (put_option put) (sample_option sample)"
simps_1: fixes x2 :: "t"   and y2 :: "t" shows "(TSome x2 = TSome y2) = (x2 = y2)"
finite_almost_full_on: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A"   and "reflp_on P A" shows "almost_full_on P A"
fw_step_Suc: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "\<forall>k'\<le>n. \<one> \<le> fw m n k n n k' k'"   and "i \<le> n"   and "j \<le> n"   and "Suc k \<le> n" shows "fw m n (Suc k) i j i j = min (fw m n k n n i j) (fw m n k n n i (Suc k) + fw m n k n n (Suc k) j)"
lideal_def: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "lideal I R addition multiplication zero unit \<equiv> subgroup_of_additive_group_of_ring I R addition multiplication zero unit \<and> lideal_axioms I R multiplication"
oalist_inv_raw_update_by_fun_pair: fixes xs :: "('a \<times> 'c) list"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv_raw xs" shows "oalist_inv_raw (update_by_fun_pair k f xs)"
fmap_def: shows "fmap = COERCE((udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'g \<rightarrow> udom\<cdot>'g, ('d \<rightarrow> 'e) \<rightarrow> 'd\<cdot>'g \<rightarrow> 'e\<cdot>'g)\<cdot>fmapU"
axioms_1: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "dlat le" shows "dpo le"
sp_instr: shows "Subx.sp_instr (Fubx_get F) (Subx.generalize_instr (body fd ! pc)) (None # map Map.empty \<Sigma>2) = Ok (None # map (typeof \<circ> Subx.box_operand) \<Sigma>2)"
HK_2: shows "hom_boundary (p + 1) X S \<in> Group.iso (relative_homology_group (p + 1) X S) (subgroup_generated (homology_group p (subtopology X S)) H)"
not_eq_order_implies_strict: fixes a :: "'c list"   and b :: "'c list" assumes "a \<noteq> b"   and "suffix b a" shows "strict_suffix b a"
wf_Int1: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "wf r" shows "wf (r \<inter> r')"
intros: fixes operations :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list"   and \<tau> :: "'a"   and name :: "String.literal"   and \<pi> :: "'a list"   and oper' :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option"   and oper :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" assumes "static_operation' operations \<tau> name \<pi> oper'"   and "oper \<noteq> oper'" shows "static_operation_not_unique operations \<tau> name \<pi> oper"
stern_brocot_rationals_not_repeated: fixes path :: "dir list"   and path' :: "dir list" assumes "root (traverse_tree path (pure rat_of \<diamondop> stern_brocot_recurse)) = root (traverse_tree path' (pure rat_of \<diamondop> stern_brocot_recurse))" shows "path = path'"
linear_imp_linear_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "linear f"   and "subspace A"   and "subspace B" shows "linear_on A B (*\<^sub>R) (*\<^sub>R) f"
comp_reduce: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and k :: "'a \<times> 'a"   and h :: "'a \<times> 'a" assumes "VV.comp f g = k"   and "VV.seq f g"   and "VV.seq g h" shows "VV.comp f (VV.comp g h) = VV.comp k h"
inj_Pair_2: fixes c :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "inj_on (\<lambda>x. (c x, x)) S"
pickT_pred_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and II :: "nat set"   and n :: "nat" shows "pickT_pred cl P II n \<equiv> n < length cl \<and> II \<in> local.shift cl n ` P n"
convex_on: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "convex S" shows "convex_on S f = (\<forall>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow> f (\<Sum>i = 1..k. u i *\<^sub>R x i) \<le> (\<Sum>i = 1..k. u i * f (x i)))"
sort_coeff_def: fixes ys :: "char list list" shows "sort_coeff ys = SPEC (\<lambda>xs. mset xs = mset ys \<and> sorted_wrt (rel2p (Id \<union> var_order_rel)) xs)"
zcf_to_poly: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R" shows "zcf (to_polynomial R a) = a"
check_sat_terminates': fixes s :: "('i, 'a) state"   and s\<^sub>0 :: "('i, 'a) state" assumes "\<nabla> s"   and "\<triangle> (\<T> s)"   and "\<diamond> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "(s\<^sub>0, s) \<in> {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}\<^sup>*" shows "check_dom s"
fps_shift_times_fps_X: fixes f :: "'a fps" assumes "1 \<le> subdegree f" shows "fps_shift 1 f * fps_X = f"
elims_2: fixes x :: "iface"   and xa :: "iface" assumes "x \<le> xa" obtains uu_ :: "char list" where "x = Iface []" and "xa = Iface uu_"   | a :: "char" and as :: "char list" and b :: "char" and bs :: "char list" where "x = Iface (a # as)" and "xa = Iface (b # bs)" and "if a = b then Iface as \<le> Iface bs else a \<le> b"
map_sel_1: fixes a :: "'b llist"   and f :: "'b \<Rightarrow> 'c" assumes "\<not> lnull a" shows "lhd (lmap f a) = f (lhd a)"
inv_map_total: fixes \<pi> :: "'d \<Rightarrow> 'c" assumes "surj \<pi>" shows "inv_map (Some \<circ> \<pi>) = Some \<circ> inv \<pi>"
exp_at_bot: shows "(exp \<longlongrightarrow> 0) at_bot"
simps_14: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (Imp p q) = (Ifm bs p \<longrightarrow> Ifm bs q)"
exec_impl_steps: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> (c # cs, css, s) \<rightarrow>\<^sup>* (cs, css, t)"
is_iso_of_sheaves: shows "iso_sheaves_of_rings Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf \<phi>\<^sub>f"
trans: fixes a :: "'e"   and b :: "'e"   and c :: "'e" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
proc_ord2a: fixes P :: "'b process"   and Q :: "'b process"   and s :: "'b event list"   and X :: "'b event set" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "((s, X) \<in> F P) = ((s, X) \<in> F Q)"
rb_aux_inv_preserved_3_1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "rb_aux_inv (bs, ss, p # ps)"   and "\<not> sig_crit bs (new_syz_sigs ss bs p) p"   and "rep_list (sig_trd bs (poly_of_pair p)) \<noteq> 0" shows "rb_aux_inv (sig_trd bs (poly_of_pair p) # bs, new_syz_sigs ss bs p, add_spairs ps bs (sig_trd bs (poly_of_pair p)))"
desargues_config_not_col_10: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col B C C'"
flip: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_bilinear prod" shows "bounded_bilinear (\<lambda>x y. prod y x)"
I_2: shows "indep_vars (\<lambda>_. borel) X I"
seqSubstSum: fixes P :: "pi"   and Q :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "(P \<oplus> Q)[<\<sigma>>] = P[<\<sigma>>] \<oplus> Q[<\<sigma>>]"
ran_map_option: fixes f :: "'c \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option" shows "ran (\<lambda>x. map_option f (m x)) = f ` ran m"
simps_6: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "params (Impl p q) = params p \<union> params q"
mult: fixes a :: "nat"   and b :: "nat" shows "fds_nth f (a * b) = fds_nth f a * fds_nth f b"
lspasl_d_der: fixes h1 :: "'a"   and h2 :: "'a" assumes "h1,h1\<triangleright>h2" shows "(0::'a,0::'a\<triangleright>0::'a) \<and> h1 = (0::'a) \<and> h2 = (0::'a)"
delta: fixes S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'd" assumes "finite S" shows "aggregation.sum_0 (\<lambda>k. if k = a then b k else zero) S = (if a \<in> S then b a + zero else zero)"
af_subsequence_U: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and n :: "nat" shows "af (\<phi> U\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) or\<^sub>n (af \<phi> (prefix (Suc n) w) and\<^sub>n af (\<phi> U\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n]))"
reads_consistent_mem_eq_on_non_volatile_reads: fixes A :: "nat set"   and m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and pending_write :: "bool"   and \<O> :: "nat set" assumes "\<forall>a\<in>A. m' a = m a"   and "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb \<subseteq> A"   and "reads_consistent pending_write \<O> m sb" shows "reads_consistent pending_write \<O> m' sb"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>A. g x = True" shows "F g A = True"
State_Idle_append_replicate_NoMsg_output_last_message: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'g message_af list"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun xs c))" shows "last_message (map output_fun (f_Exec_Comp_Stream trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c)) = last_message (map output_fun (f_Exec_Comp_Stream trans_fun xs c))"
SUP_union: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and M :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set"   and B :: "'c::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Sup::'b::type set \<Rightarrow> 'b::type) ((M::'c::type \<Rightarrow> 'b::type) ` ((A::'c::type set) \<union> (B::'c::type set))) = (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (Sup (M ` A)) (Sup (M ` B))"
ring_simprules_1: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
llist_set_induct: fixes x :: "'a"   and xs :: "'a llist"   and P :: "'a \<Rightarrow> 'a llist \<Rightarrow> bool" assumes "x \<in> lset xs"   and "\<And>xs. \<not> lnull xs \<Longrightarrow> P (lhd xs) xs"   and "\<And>xs y. \<lbrakk>\<not> lnull xs; y \<in> lset (ltl xs); P y (ltl xs)\<rbrakk> \<Longrightarrow> P y xs" shows "P x xs"
injective_imp_proper_eq_closed_map: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c topology"   and Y :: "'d topology" assumes "inj_on f (topspace X)" shows "proper_map X Y f = closed_map X Y f"
del_bal: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and t :: "'b btree"   and x :: "'b" assumes "split split"   and "0 < k"   and "root_order k t"   and "bal t" shows "bal (split.del split k x t)"
13: shows "\<exists>i. f (Suc (nat (padic_val p f + padic_val p g))) = i * p ^ nat (padic_val p f) \<and> \<not> p dvd int (nat i)"
get_ancestors_ptr_in_heap: fixes get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_get_ancestors get_ancestors"   and "h \<turnstile> ok get_ancestors ptr" shows "ptr |\<in>| object_ptr_kinds h"
a'_def: shows "a' = (\<lambda>i. if i = \<lbrakk>new_id\<rbrakk> then \<lbrakk>\<Psi> (a (P m'), a (S m')) new_id\<rbrakk> else a i)"
lpeak_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and xs :: "(accent \<times> 'b) list" shows "local.lpeak r a b xs = (\<exists>as b' cs a' bs. snd ` set as \<subseteq> under r a \<and> b' \<in> {[(Grave, b)], []} \<and> snd ` set cs \<subseteq> under r a \<union> under r b \<and> a' \<in> {[(Acute, a)], []} \<and> snd ` set bs \<subseteq> under r b \<and> xs = as @ b' @ cs @ a' @ bs)"
span_breakdown_eq: fixes x :: "'b"   and a :: "'b"   and S :: "'b set" shows "(x \<in> cspan (insert a S)) = (\<exists>k. x - k *\<^sub>C a \<in> cspan S)"
nn_integral_fst: fixes M :: "'c measure"   and f :: "'d \<times> 'c \<Rightarrow> ennreal"   and M1 :: "'d measure" assumes "sigma_finite_measure M"   and "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f"
val_ifex: fixes b :: "'c bexp"   and s :: "'c \<Rightarrow> bool" shows "val_ifex (Boolean_Expression_Example.ifex_of b) s = bval b s"
abelian_subgroup_axioms: shows "abelian_subgroup H G"
simps_1: fixes f :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "transfer \<Up>f s = f s"
3: shows "\<turnstile> DBSpec \<longrightarrow> WF(Deq inp q1 mid)_vars"
sup_continuous_def: fixes F :: "'d \<Rightarrow> 'e" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (\<Sqinter>range M) = (\<Sqinter>i. F (M i)))"
max_wp_DC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "maximal (wp a)"   and "maximal (wp b)" shows "maximal (wp (a \<Sqinter> b))"
vector_scalar_matrix_ac: fixes x :: "('b, 'd) vec"   and k :: "'b"   and A :: "(('b, 'c) vec, 'd) vec" shows "x v* (k *k A) = k *s (x v* A)"
bi_unique_alt_def: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
Abs_poly_mapping_inverse: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'c)}}" shows "poly_mapping.lookup (Abs_poly_mapping y) = y"
analyzed_closed_mod_timpls_def: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and TI :: "('sets set \<times> 'sets set) list" shows "analyzed_closed_mod_timpls M TI \<equiv> let f = list_all (intruder_synth_mod_timpls M TI); g = \<lambda>t. if f (fst (Ana t)) then f (snd (Ana t)) else \<forall>s\<in>comp_timpl_closure {t} (set TI). case Ana s of (K, R) \<Rightarrow> f K \<longrightarrow> f R in list_all g M"
less: fixes y :: "'a tree list" assumes "(sum_list \<circ> map size) y < (sum_list \<circ> map size) ts" shows "T_list_fast_rec y \<le> (\<Sum>t\<leftarrow>y. 7 * size t + 1)"
prems_2: shows "R_lin_independent ((m # ms) @ n # ns)"
addpreRT_welldefined: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:16..PRreq-:18} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRreq-:17 \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRrep-:5 \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRrep-:6 \<longrightarrow> the (nhop (rt \<xi>) (dip \<xi>)) \<in> kD (rt \<xi>)))"
lb_arctan: fixes k :: "int"   and prec :: "nat"   and na :: "nat" assumes "1 < k" shows "real_of_float (lapprox_rat prec 1 k * lb_arctan_horner prec (get_even na) 1 (float_round_up (Suc prec) (lapprox_rat prec 1 k * lapprox_rat prec 1 k))) \<le> arctan (1 / real_of_int k)"
that: fixes e' :: "Syntax.term"   and eV' :: "Syntax.term" assumes "{$$} \<turnstile> e', eP', eV' : \<tau> \<and> \<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater> \<and> \<lless>\<pi>, eV\<ggreater> V\<rightarrow> \<lless>[], eV'\<ggreater>" shows "thesis"
these_empty_eq: fixes B :: "'a option set" shows "(Option.these B = {}) = (B = {} \<or> B = {None})"
arctan_float_intervalI: fixes x :: "real"   and X :: "float interval"   and p :: "nat" assumes "x \<in>\<^sub>r X" shows "arctan x \<in>\<^sub>r arctan_float_interval p X"
INF_lower: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "i \<in> A" shows "less_eq_prod (Inf_prod (f ` A)) (f i)"
alls2_ne: fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "alls2 A B a b \<noteq> []"
minimal_imp_inductive_on: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>Q x. x \<in> Q \<and> Q \<subseteq> A \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. P y z \<longrightarrow> y \<notin> Q)" shows "inductive_on P A"
WHILET_rule: fixes R :: "('a \<times> 'a) set"   and I :: "'a \<Rightarrow> bool"   and s :: "'a"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "wf R"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> (spec s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s\<rbrakk> \<Longrightarrow> \<Phi> s" shows "while\<^sub>T b f s \<le> SPEC \<Phi>"
ranI: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a" assumes "m a = \<lfloor>b\<rfloor>" shows "b \<in> ran m"
N_def: shows "N = (\<lambda>j. if j < dg then coeff_int f (int k - int (dg - 1) + int j) else coeff_int g (int k - int (df - 1) + int (j - dg)))"
satA_o: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and at :: "('fsym, 'psym) atm" shows "satA (int \<xi> \<circ> \<rho>) at = satA \<xi> (substA \<rho> at)"
order_top_ow_axioms: shows "order_top_ow U\<^sub>1 (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>1) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>1) top\<^sub>1"
simps_5: fixes uu :: "nat"   and v :: "name" shows "wellformed' uu (Free v) = True"
Methd_triple_valid2_0: fixes G :: "prog"   and P :: "'b assn"   and C :: "qtname"   and sig :: "sig"   and Q :: "'b assn" shows "G\<Turnstile>0\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
simps: fixes G :: "'b list set"   and us :: "'b list list" shows "Ref G us = concat (map (decompose G) us)"
cases: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'d" assumes "(r OO s) a1 a2" obtains a :: "'b" and b :: "'c" and c :: "'d" where "a1 = a" and "a2 = c" and "r a b" and "s b c"
induct: fixes P :: "'a \<Rightarrow> 'a tree \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a tree" assumes "\<And>uu_. P uu_ \<langle>\<rangle>"   and "\<And>x l y r. \<lbrakk>y < x \<Longrightarrow> P x r; \<lbrakk>\<not> y < x; x < y\<rbrakk> \<Longrightarrow> P x l\<rbrakk> \<Longrightarrow> P x \<langle>l, y, r\<rangle>" shows "P a0 a1"
nonsat_potential_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> 'b" shows "nonsat_potential f l \<equiv> sum l {v \<in> V. (0::'a) < excess f v}"
widen_array_object: fixes P :: "'a prog"   and A :: "ty" shows "P \<turnstile> A\<lfloor>\<rceil> \<le> Class Object"