hcju/isabelleps huggingface dataset:
imageI: fixes x :: "'f"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'g" assumes "x \<in> A" shows "f x \<in> f ` A"
ext_irrefl_before_trans_axioms: shows "ext_irrefl_before_trans len_lexext"
two_comparisons_into_case_order_2: fixes x :: "'a"   and y :: "'a"   and P :: "'d"   and Q :: "'d"   and R :: "'d" shows "(if le x y then if y = x then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
section_is_mono: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "section g" shows "local.mono g"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRelLPO Enc"
inverse_inverse: fixes U :: "'a set"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group_add_ow U minus plus zero uminus"   and "a \<in> U" shows "uminus (uminus a) = a"
atLeastLessThan_subseteq_atLeastAtMost_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.atLeastLessThan less_eq less a b \<subseteq> ord.atLeastAtMost less_eq c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
extremum_uniqueI: fixes a :: "nat" assumes "a \<le> 0" shows "a = 0"
nsum_zeroGTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. f j = \<zero>) \<longrightarrow> \<Sigma>\<^sub>e A f n = \<zero>"
dghm_singleton_ArrMap_app: fixes x :: "V"   and \<CC> :: "V"   and j :: "V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "dghm_singleton j \<CC>\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = ZFC_in_HOL.set {\<langle>j, x\<rangle>}"
arr_cod: fixes f :: "'s" assumes "S.arr f" shows "S.arr (S.cod f)"
wfP_app: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a" assumes "wfP p" shows "wfP (\<lambda>x y. p (f x) (f y))"
ds_subid_eq: fixes x :: "'a" shows "(x \<sqsubseteq> (1::'a)) = (x = d x)"
intensional_simps_23: fixes P :: "'b \<Rightarrow> bool" shows "P \<or> #True \<equiv> #True"
assms_3: shows "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>'\<^esup> n'"
kG: fixes x :: "'b" shows "norm (g' x) \<le> norm x * kG"
xivp_1: shows " D X = (\<lambda>t. f (X t)) on {0--t}"
orecvmsg_simps_4: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and ip :: "nat" shows "orecvmsg P \<sigma> (\<not>unicast ip) = True"
fst_edge_at: fixes c :: "'form"   and "is" :: "nat list" shows "fst (edge_at c is) = edge_from c is"
mem_conc_forms: fixes a :: "'form" shows "(a |\<in>| conc_forms) = (a \<in> set conclusions)"
simple_path_cases: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "arc g \<or> pathfinish g = pathstart g"
degree_i_def: fixes pp :: "'b list" shows "degree_i pp \<equiv> length pp - 1"
inner_zero_left: fixes x :: "'a" shows "(0::'a) \<bullet> x = 0"
A_authenticates_B: fixes K :: "nat"   and Ta :: "nat"   and evs :: "event list"   and A :: "agent"   and Tk :: "nat"   and B :: "agent"   and X :: "msg" assumes "Crypt K (Number Ta) \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Says B A (Crypt K (Number Ta)) \<in> set evs"
inf': shows "\<forall>i. \<Gamma>\<turnstile>Call (p i) \<down> Normal (s i) \<and> (\<exists>c. \<Gamma>\<turnstile> (Call (p i), Normal (s i)) \<rightarrow>\<^sup>+ (c, Normal (s (i + 1))) \<and> redex c = Call (p (i + 1)))"
zero_order_1: fixes x :: "'i multiset" shows "{#} \<subseteq># x"
sub_of_denot_equiv_ground': fixes E :: "char list \<Rightarrow> hterm"   and G :: "char list \<Rightarrow> hterm list \<Rightarrow> bool"   and l :: "fterm literal" shows "eval\<^sub>l E HFun G l = eval\<^sub>l E HFun G (l \<cdot>\<^sub>l sub_of_denot E) \<and> ground\<^sub>l (l \<cdot>\<^sub>l sub_of_denot E)"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_backto_standard_pos_B ires rs (l, r) = (\<exists>ln rn. l = Bk # Bk \<up> ln @ Oc # ires \<and> r = Bk # Oc \<up> Suc (Suc rs) @ Bk \<up> rn)"
bl_to_bin_def: fixes bs :: "bool list" shows "bl_to_bin bs = bl_to_bin_aux bs 0"
contract_vacuity: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<phi> \<notin> K" shows "K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi> = K"
LI_in_measure: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto> (S\<^sub>2, \<theta>\<^sub>2)" shows "((S\<^sub>2, \<theta>\<^sub>2), S\<^sub>1, \<theta>\<^sub>1) \<in> measure\<^sub>s\<^sub>t"
card_UNIV_finite_3_def: shows "card_UNIV_class.card_UNIV = Phantom(Enum.finite_3) 3"
inject: fixes x1 :: "'d"   and y1 :: "'d" shows "(ln x1 = ln y1) = (x1 = y1)"
meet_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
fM_rel_def: shows "fM_rel = {(in_sync, hp, fM, fM, [], l) |fM hp in_sync l. hp = hp_Idle \<longrightarrow> \<not> in_sync} \<union> {(in_sync, hp_Idle, fM, fM', [], l) |fM fM' in_sync l. in_sync} \<union> {(in_sync, hp_Idle, \<not> fM, fM, [mw_fM (\<not> fM)], False) |fM in_sync. in_sync}"
delete_notin_dom: fixes k :: "'c"   and al :: "('c \<times> 'd) list" shows "k \<notin> fst ` set (AList.delete k al)"
join_def: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y = (THE sup. is_sup x y sup)"
span_add_eq2: fixes y :: "'b"   and S :: "'b set"   and x :: "'b" assumes "y \<in> ideal S" shows "(x + y \<in> ideal S) = (x \<in> ideal S)"
linear_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and f :: "'f \<Rightarrow> 'h" shows "linear_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 f \<equiv> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<and> module_hom_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 f"
uniform_discrete_def: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.metric_space dist uniformity open" shows "metric_space.uniform_discrete dist S = (\<exists>e>0. \<forall>x\<in>S. \<forall>y\<in>S. dist x y < e \<longrightarrow> x = y)"
inf_continuousD: fixes F :: "'a \<Rightarrow> 'b"   and M :: "nat \<Rightarrow> 'a" assumes "inf_continuous F"   and "decseq M" shows "F (\<Sqinter> range M) = (\<Sqinter>i. F (M i))"
substAll_2: fixes P :: "('node \<Rightarrow> 'val set) \<times> ('node \<times> 'val \<Rightarrow> 'val list option) \<Rightarrow> bool"   and g :: "'g"   and Q :: "('node \<Rightarrow> 'val set) \<Rightarrow> ('node \<times> 'val \<Rightarrow> 'val list option) \<Rightarrow> bool" assumes "P (uses g, phis g)"   and "\<And>x. \<lbrakk>P x; uninst.inst' uses phis g x; cond g x\<rbrakk> \<Longrightarrow> P (uninst.step uses phis g x)"   and "\<And>x. \<lbrakk>P x; uninst.inst' uses phis g x; \<not> cond g x\<rbrakk> \<Longrightarrow> Q (fst x) (snd x)" shows "Q (uninst.uses'_all uses phis g) (uninst.phis'_all uses phis g)"
reducedly_statically_complete_calculus_axioms_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" shows "reducedly_statically_complete_calculus_axioms Bot Inf entails Red_I Red_F \<equiv> \<forall>B N. B \<in> Bot \<longrightarrow> calculus.reduc_saturated Inf Red_I Red_F N \<longrightarrow> entails N {B} \<longrightarrow> (\<exists>B'\<in>Bot. B' \<in> N)"
sub2_grows_triangle: fixes A :: "'a mat"   and l :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat" assumes "det_selection_fun sel_fun"   and "mute_fun mf"   and "local.sub2 (dim_row A) l (r, A) = (r', A')"   and "local.triangular_to l A"   and "l < dim_col A"   and "l < dim_row A" shows "local.triangular_to (Suc l) A'"
e_tuple_nth_le: fixes n :: "nat"   and a :: "nat"   and i :: "nat" assumes "n \<le> a" shows "e_tuple_nth a i n = tuple_decode a i ! n"
toQ_cases: fixes x :: "'a['c, 'b]" obtains y :: "('a, sdim, 'b) Measurement_System" where "x = toQ y" and "y \<in> {x. Quantity.dim x = QD('c)}"
prefix_not_empty: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix xs ys"   and "xs \<noteq> []" shows "ys \<noteq> []"
squarefree_part_nonzero: fixes n :: "'a" shows "squarefree_part n \<noteq> (0::'a)"
distinct_4: fixes x31 :: "char list"   and x32 :: "char list list"   and x1 :: "instr" shows "StepC2 x31 x32 \<noteq> StepI x1"
makes_compatible_eval_sched: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and sched :: "nat list"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"   and "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"   and "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>1', mem\<^sub>1')" shows "\<exists>cms\<^sub>2' mem\<^sub>2' mems'. sound_mode_use (cms\<^sub>1', mem\<^sub>1') \<and> sound_mode_use (cms\<^sub>2', mem\<^sub>2') \<and> map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> (cms\<^sub>2, mem\<^sub>2) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>2', mem\<^sub>2') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'"
signed_take_bit_minus: fixes n :: "nat"   and k :: "int" shows "signed_take_bit n (- signed_take_bit n k) = signed_take_bit n (- k)"
lexordp_append_left_rightI: fixes x :: "'a"   and y :: "'a"   and us :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "x < y" shows "lexordp (us @ x # xs) (us @ y # ys)"
assms: shows "long_ch_by_ord f X \<and> infinite X"
k: shows "clock_simAbs k \<in> (\<Union>a. Clock.sim_equiv_class a ` Clock.jkbpC)"
gen_ramanujan_sum_distrib: fixes a :: "nat"   and b :: "nat"   and m :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> complex"   and g :: "nat \<Rightarrow> complex" assumes "0 < a"   and "0 < b"   and "0 < m"   and "0 < k"   and "coprime a k"   and "coprime b m"   and "coprime k m"   and "multiplicative_function f"   and "multiplicative_function g" shows "s f g (m * k) (a * b) = s f g m a * s f g k b"
dom_dom: fixes f :: "'a" assumes "C.arr f" shows "C.dom (C.dom f) = C.dom f"
less_irrefl: fixes x :: "'x \<Rightarrow>\<^sub>0 nat" shows "\<not> x \<prec> x"
payload_Sign: fixes X :: "msg"   and Y :: "msg" assumes "Sign X Y \<in> payload" obtains "X \<in> payload" and "Y \<in> payload"
Sup_V_def: fixes X :: "V set" shows "\<Squnion> X \<equiv> if small X then set (\<Union> (elts ` X)) else 0"
p'_def: fixes t :: "real" shows "delayed_safe_distance.other.p' t = other.s' \<delta> + a\<^sub>o * t"
doms_eqvt_1: fixes pi :: "(tyvrs \<times> tyvrs) list"   and \<Gamma> :: "binding list" shows "pi \<bullet> ty_dom \<Gamma> = ty_dom (pi \<bullet> \<Gamma>)"
gSwapCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSwapCls MOD \<equiv> gSwapGVar MOD \<and> gSwapGAbs MOD \<and> gSwapGOp MOD"
absorb_iff2: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subseteq># b) = (a \<union># b = b)"
take_Nil: fixes n :: "nat" shows "take n \<epsilon> = \<epsilon>"
trans: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
ground_subst_ground_cls: fixes \<sigma> :: "'s"   and C :: "'a literal multiset" assumes "is_ground_subst \<sigma>" shows "is_ground_cls (C \<cdot> \<sigma>)"
trapG1_def: shows "trapG1 = pos_quad \<inter> ({p. snd p - 751 / 100 \<le> 0} \<inter> {p. fst p + snd p - 812 / 100 \<le> 0})"
fg_genTr0: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and z :: "'d"   and x :: "'d" assumes "fgmodule_condition R f i s A z"   and "x \<in> aug_pm_set z i (s_set R s A)" shows "x \<in> linear_span R (fgmodule R A z i f s) (carrier R) A"
B: shows "B \<in> smallest_ccdi_sets \<Omega> M"
DynCastRedsFail: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s' :: "(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 D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s'\<rangle>"   and "hp s' a = \<lfloor>(D, S)\<rfloor>"   and "\<not> P \<turnstile> Path D to C unique"   and "\<not> P \<turnstile> Path last Cs to C unique"   and "C \<notin> set Cs" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>"
exists: shows "\<exists>i. A $ i $ j = (1::'a) \<and> j = (LEAST n. A $ i $ n \<noteq> (0::'a))"
pop_aassoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "a \<plusminus> b \<plusminus> c = a \<plusminus> (b \<plusminus> c)"
Hermite_ring_imp_Bezout_ring: assumes "OFCLASS('b, Hermite_ring_class)" shows "OFCLASS('b, bezout_ring_class)"
OclOr_commute: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X or Y) = (Y or X)"
cffi: shows "f O f\<inverse> \<subseteq> e \<union> f \<union> f\<inverse>"
compact_lemma_general: fixes basis :: "'a set"   and proj :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'c"   and unproj :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c" assumes "finite basis"   and "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. proj x k) ` range f)"   and "\<And>e k. k \<in> basis \<Longrightarrow> proj (unproj e) k = e k"   and "\<And>x. unproj (proj x) = x" shows "\<forall>d\<subseteq>basis. \<exists>l r. strict_mono r \<and> (\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (proj (f (r n)) i) (proj l i) < e)"
F4: shows "\<exists>u SKF\<^sub>7. \<bar>g (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<and> (0::'a) \<le> \<bar>inverse c\<bar>"
ssubst_Obj: fixes z :: "char list"   and u :: "sterm"   and f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type" shows "[z \<rightarrow> u] Obj f T = Obj (\<lambda>l. ssubst_option z u (f l)) T"
subset_insert: fixes x :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "x \<notin> A" shows "(A \<subseteq> insert x B) = (A \<subseteq> B)"
interaction_bounded_by'_exec_gpv_count: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'b) gpv"   and n :: "nat"   and x :: "'e"   and s' :: "'d \<Rightarrow> 'b option"   and s :: "'d \<Rightarrow> 'b option"   and count :: "('d \<Rightarrow> 'b option) \<Rightarrow> nat" assumes "interaction_bounded_by' consider gpv n"   and "(x, s') \<in> set_spmf (exec_gpv random_permutation gpv s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (random_permutation s x); finite (dom s); consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (random_permutation s x); finite (dom s); \<not> consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "finite (dom s)" shows "count s' \<le> n + count s"
viterbi_valid_path: fixes s :: "'s"   and t_end :: "'s"   and os :: "'t list"   and as :: "'s list"   and v :: "ennreal" assumes "viterbi s t_end os = (as, v)" shows "length as = length os \<and> set as \<subseteq> \<S>"
fun_upd_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'c" shows "(f \<circ>\<circ>\<circ> fun_upd g) x y = (f \<circ> g)(x := f y)"
\<dd>_def: fixes language :: "'a language" shows "\<dd> language = (case language of Lang x1 x2 \<Rightarrow> x2)"
assms: shows "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"
Tm: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "mpt M T" shows "T \<in> measure_preserving M M"
lins2D_2: shows "set (Inl v # \<pi>) = (fset G^V - fset S) <+> set G^E"
mono_dp_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and w :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit"   and "relation_algebra.is_direct_product inf top composition converse unit w x"   and "relation_algebra.is_direct_product inf top composition converse unit y z" shows "composition (relation_algebra.Phi inf composition converse w x y z) y = w"
simps_2: fixes x :: "'a"   and t :: "'a tree \<times> nat \<times> nat" shows "nxt (Delete x) t = local.delete2 x t"
g_has_sgnx2: assumes "g st = 0"   and "0 < sin st \<or> cos st = 1" shows "(g has_sgnx - 1) (at_right st)"
bmod_oner: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a" shows "\<otimes> \<rho> \<delta> f x = f x"
division_decomp: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a (times b c)" shows "\<exists>b' c'. a = times b' c' \<and> dvd.dvd times b' b \<and> dvd.dvd times c' c"
uncurry_apply: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "uncurry f (a, b) = f a b"
bind_spmf_cong_simp: fixes p :: "'a spmf"   and q :: "'a spmf"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'b spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q =simp=> f x = g x" shows "p \<bind> f = q \<bind> g"
transpose_transpose: fixes A :: "'b mat" shows "A\<^sup>T\<^sup>T = A"
cswitch_consistency_and_respect: fixes s :: "state_t"   and u :: "partition_id_t"   and new_current :: "thread_id_t" assumes "atomic_step_invariant s" shows "vpeq u s (s\<lparr>current := new_current\<rparr>)"
l_to_tree_order_wf_axioms: shows "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes"
ins_list_split: fixes ts :: "('a btree \<times> 'a) list"   and x :: "'a"   and ls :: "('a btree \<times> 'a) list"   and rs :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "split ts x = (ls, rs)"   and "sorted_less (inorder (Node ts t))" shows "ins_list x (inorder (Node ts t)) = inorder_list ls @ ins_list x (inorder_list rs @ inorder t)"
mult_strict_right_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times a c) (times b c)"
power2_minus: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus a) 2 = power.power one times a 2"
arrE: fixes f :: "'a \<times> 'a" assumes "CC.arr f" obtains "C.arr (fst f) \<and> C.arr (snd f)"
\<xi>: shows "[\<lparr>A!,b\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,b\<^sup>P\<rparr>) \<^bold>& \<^bold>\<not>\<lbrace>b\<^sup>P,\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>\<rbrace> in v]"
HaddP_OrdP: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and z :: "tm" assumes "H \<turnstile> HaddP x y z"   and "H \<turnstile> OrdP x" shows "H \<turnstile> OrdP z"
chine_eq_apex: shows "\<pi>.chine = \<pi>.apex"