hcju/isabelleps huggingface dataset:
not_in_butlast: fixes x :: "'b"   and ys :: "'b list" assumes "x \<in> set ys"   and "x \<notin> set (butlast ys)" shows "x = last ys"
map_reverse_iterateoi_def: fixes \<alpha> :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and invar :: "'c \<Rightarrow> bool"   and reverse_iterateoi :: "'c \<Rightarrow> ('f \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f" shows "map_reverse_iterateoi \<alpha> invar reverse_iterateoi \<equiv> ordered_finite_map \<alpha> invar \<and> map_reverse_iterateoi_axioms \<alpha> invar reverse_iterateoi"
Says_ticket_parts: fixes S :: "agent"   and A :: "agent"   and K :: "nat"   and SesKey :: "msg"   and B :: "msg"   and TimeStamp :: "msg"   and Ticket :: "msg"   and evs :: "event list" assumes "Says S A (Crypt K \<lbrace>SesKey, B, TimeStamp, Ticket\<rbrace>) \<in> set evs" shows "Ticket \<in> parts (knows Spy evs)"
in_set_vwalk_arcs_append1: fixes x :: "'a \<times> 'a"   and p :: "'a list"   and q :: "'a list" assumes "x \<in> set (vwalk_arcs p) \<or> x \<in> set (vwalk_arcs q)" shows "x \<in> set (vwalk_arcs (p @ q))"
nat_term_compD3: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a"   and t :: "'a" assumes "nat_term_comp cmp"   and "cmp u v = Lt" shows "cmp (splus t u) (splus t v) = Lt"
divides_pirreducible_condition: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set"   and q :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list"   and p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "pirreducible\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub> K q"   and "p \<in> carrier (K [X]\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub>)"   and "p divides\<^bsub>K [X]\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub>\<^esub> q" shows "p \<in> Units (K [X]\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub>) \<or> p \<sim>\<^bsub>K [X]\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub>\<^esub> q"
normalize_rules_dnf_append: fixes rs1 :: "'b rule list"   and rs2 :: "'b rule list" shows "normalize_rules_dnf (rs1 @ rs2) = normalize_rules_dnf rs1 @ normalize_rules_dnf rs2"
Collect_case_prodD: fixes x :: "'g \<times> 'h"   and A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
power_minus_mult: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.monoid_mult one times"   and "0 < n" shows "times (power.power one times a (n - 1)) a = power.power one times a n"
csplit_def: shows "csplit = (\<Lambda> f p. f\<cdot>(fst p)\<cdot>(snd p))"
ifex_sat_SomeD: fixes i :: "'a ifex"   and ass :: "'a \<Rightarrow> bool" assumes "ifex_no_twice i"   and "ifex_sat i = Some ass" shows "val_ifex i ass = True"
chainTransitionAppend_4: fixes P :: "pi"   and P'' :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P''"   and "P'' \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<Longrightarrow>\<alpha> \<prec> P'"
not_singleton_card: assumes "class.not_singleton TYPE('c)" shows "\<exists>x y. x \<noteq> y"
eq_fin_remove1: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and c :: "'item" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "remove1 c w\<^sub>1 =\<^sub>F remove1 c w\<^sub>2"
pause_option_def: fixes pause :: "'f \<Rightarrow> ('g \<Rightarrow> 'e) \<Rightarrow> 'e"   and out :: "'f"   and c :: "'g \<Rightarrow> ('d, 'e) optionT" shows "pause_option pause out c = OptionT (pause out (\<lambda>i. run_option (c i)))"
Max_wrt_prefs_nonempty: fixes R :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "total_preorder_on alts R" shows "Max_wrt R \<noteq> {}"
ord_resolve_13: shows "\<forall>i<n_twin. S_M S M (CAs ! i) = {#}"
prems_1: shows "steps0 (st, l, r) ap (Suc stp) = (st', l', r')"
compute_lex_pp: fixes p :: "('a, 'b) pp"   and q :: "('a, 'b) pp" shows "lex_pp p q = (lex_comp' p q \<noteq> Gt)"
inv_insert: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t :: "('b \<times> 'c) tree"   and x :: "'b" assumes "Set2_Join join inv"   and "inv t" shows "inv (Set2_Join.insert join x t)"
simps_6: fixes cl :: "('a, 'b, 'c) cmd list" shows "finished (Par cl) = Ball (set cl) finished"
disc_60: fixes x5 :: "protocol" shows "\<not> is_Src_Ports (Prot x5)"
cases: fixes a1 :: "'e set"   and a2 :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" assumes "(a1, a2) \<in> foldSetD D f e" obtains "a1 = {}" and "a2 = e" and "e \<in> D"   | x :: "'e" and A :: "'e set" and y :: "'f" where "a1 = insert x A" and "a2 = f x y" and "x \<notin> A" and "f x y \<in> D" and "(A, y) \<in> foldSetD D f e"
distinct_39: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm"   and x7 :: "'b ptrm" shows "PPair x51 x52 \<noteq> PSnd x7"
DOM_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "monoidal_language.Dom C (free_monoidal_category.rep C (free_monoidal_category.mkarr C t)) = monoidal_language.Dom C t"
rsp: shows "eq_onp (\<lambda>xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None) [] []"
card_seteq: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "order.greater_eq B A"   and "order.greater_eq (card A) (card B)" shows "A = B"
Sup_dual_def: shows "Sup = \<partial> \<circ> Inf \<circ> (`) \<partial>\<^sup>-"
not_produces_imp_notin_production: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset"   and A :: "'a" assumes "production N C \<noteq> {A}" shows "A \<notin> production N C"
i': shows "atom i' \<sharp> (t, X0, X1, i)"
sup_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<sqinter> y) = x"
normalize_gives_polynomial: fixes p :: "'a list list"   and K :: "'a list set" assumes "set p \<subseteq> K" shows "polynomial\<^bsub>K [X]\<^esub> K (UP.normalize p)"
cod_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.cod (src\<^sub>B \<mu>) = src\<^sub>B \<mu>"
f_def: fixes s :: "('a, 'U, 'b) state"   and s' :: "('c, 'U, 'b) state" shows "f s s' \<equiv> \<forall>u c. u \<notin> GH \<longrightarrow> (\<exists>st. s = State st u c) = (\<exists>st'. s' = State st' u c)"
IntD2: fixes c :: "'f"   and A :: "'f set"   and B :: "'f set" assumes "c \<in> A \<inter> B" shows "c \<in> B"
set_txs: shows "snd ` set txs = snd ` set (zip (map Var us) (map snd txs))"
MWLfSteps_det_def: shows "MWLfSteps_det \<equiv> {((xa, x), xc, xb). MWLfSteps_detp xa x xc xb}"
hermitean_elems_4: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "hermitean (A, B, C, D)" shows "cnj B = C"
whileAnnoFix_def: fixes b :: "'e set"   and I :: "'h \<Rightarrow> 'e set"   and V :: "'h \<Rightarrow> ('e \<times> 'e) set"   and c :: "'h \<Rightarrow> ('e, 'f, 'g) com" shows "whileAnnoFix b I V c = While b (c undefined)"
HMSet_diff: fixes A :: "hmultiset multiset"   and B :: "hmultiset multiset" shows "HMSet (A - B) = HMSet A - HMSet B"
replace_type: fixes \<Delta> :: "binding list"   and X :: "tyvrs"   and T :: "ty"   and \<Gamma> :: "binding list"   and S :: "ty" assumes "\<turnstile> (\<Delta> @ TVarB X T # \<Gamma>) ok"   and "S closed_in \<Gamma>" shows "\<turnstile> (\<Delta> @ TVarB X S # \<Gamma>) ok"
strip_comb_empty: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "snd (term.strip_comb unapp t) = []" shows "fst (term.strip_comb unapp t) = t"
HEndPhase2_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2b s"   and "HEndPhase2 s s' p" shows "Inv2b s'"
fmupd_lookup: fixes a :: "'c"   and b :: "'b"   and m :: "('c, 'b) fmap"   and a' :: "'c" shows "fmlookup (fmupd a b m) a' = (if a = a' then Some b else fmlookup m a')"
facetrel_complement_vertex: fixes y :: "'b set"   and x :: "'b set"   and v :: "'b" assumes "y \<lhd> x"   and "x = insert v y" shows "v \<notin> y"
map_add_com: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>1 ++ h\<^sub>0"
lotteries_on_def: fixes Oc :: "'b set" shows "\<P> Oc = {p. set_pmf p \<subseteq> Oc}"
Nil_notin_Lm: fixes r :: "(bool \<times> 'b) rexp" shows "[] \<notin> Lm r"
Max_ge: fixes A :: "'b DBMEntry set"   and x :: "'b DBMEntry" assumes "finite A"   and "x \<in> A" shows "x \<preceq> linordered_monoid.Max A"
left_hom_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b" shows "left_hom V H \<omega> \<equiv> weak_composition V H \<and> subcategory V (weak_composition.left V H \<omega>) \<and> left_hom_axioms V \<omega>"
comp_eq_dest_lhs: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'd"   and c :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "a \<circ> b = c" shows "a (b v) = c v"
set_gen2: fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "set (Algorithm.gen2 A B a b) = {(x, y). x \<le>\<^sub>v replicate (length a) A \<and> y \<le>\<^sub>v replicate (length b) B}"
inject: fixes x21 :: "'b"   and x22 :: "'b llist"   and y21 :: "'b"   and y22 :: "'b llist" shows "(LCons x21 x22 = LCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
induct: fixes P :: "('b \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a0 :: "('b \<times> 'b) list"   and a1 :: "'b"   and a2 :: "'b" assumes "\<And>TI a b. (\<And>x xa y. \<lbrakk>(a, b) \<notin> set TI; x \<in> set TI; (xa, y) = x\<rbrakk> \<Longrightarrow> P (removeAll (xa, y) TI) y b) \<Longrightarrow> P TI a b" shows "P a0 a1 a2"
leI: fixes y :: "'a"   and x :: "'a" assumes "\<not> y < x" shows "x \<le> y"
monad_altc_writerT: assumes "monad_altc return bind altc" shows "monad_altc local.return_writer local.bind_writer local.altc_writer"
extending_def: fixes C :: "'d program \<Rightarrow> 'd set"   and h :: "'e \<times> 'f \<Rightarrow> 'd"   and F :: "'e program"   and Y' :: "'d program set"   and Y :: "'e program set" shows "extending C h F Y' Y \<equiv> \<forall>G. extend h F ok G \<longrightarrow> F \<squnion> project h (C G) G \<in> Y \<longrightarrow> extend h F \<squnion> G \<in> Y'"
leg_simps_4: shows "C.dom Dom_g.leg1 = Dom_g.apex"
mat_diag_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "mat_diag n f = mat n n (\<lambda>(i, j). if i = j then f j else (0::'a))"
take_hd_drop: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
fds_moebius_commute: fixes x :: "'a fds" shows "x * fds moebius_mu = fds moebius_mu * x"
factor_theorem': fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "to_fun f a = \<zero>" shows "f = X_poly_minus R a \<otimes>\<^bsub>P\<^esub> UP_root_div f a"
Nil_less_conv_neq_Nil: fixes xs :: "'b list" shows "([] < xs) = (xs \<noteq> [])"
points_index_singleton_zero: fixes ps :: "'b set"   and b :: "'b set" assumes "\<not> ps \<subseteq> b" shows "{#b#} index ps = 0"
assms_4: fixes i :: "nat" shows "g i \<noteq> - \<infinity>"
retl: shows "(\<pi> @\<^bsup>l\<^esup> \<pi>l) (l + n) = local.return"
PolysI: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "keys p \<subseteq> .[X]" shows "p \<in> P[X]"
decompose: shows "set (\<Gamma> @ \<Gamma>\<^sub>r) = set \<Gamma>'"
zero_flow_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> E" shows "f (u, v) = (0::'a)"
ANR_neighborhood_retract: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "ANR U"   and "S retract_of T"   and "openin (top_of_set U) T" shows "ANR S"
induct: fixes P :: "'k \<Rightarrow> ('a, 'k) aformula \<Rightarrow> bool"   and a0 :: "'k"   and a1 :: "('a, 'k) aformula" assumes "\<And>k \<phi> \<psi>. \<lbrakk>P k \<phi>; P k \<psi>\<rbrakk> \<Longrightarrow> P k (FOr \<phi> \<psi>)"   and "\<And>k v. P k (FBool v)"   and "\<And>k v. P k (FBase v)"   and "\<And>k v. P k (FNot v)"   and "\<And>k v va. P k (FAnd v va)"   and "\<And>k v va. P k (FEx v va)"   and "\<And>k v va. P k (FAll v va)" shows "P a0 a1"
presOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'g, 'opSym, 'var, 'e, 'f, 'h) Recursion.model_scheme" shows "presOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = gOp MOD delta inp (lift h inp) binp (lift hA binp)"
zero_less_Suc: fixes n :: "nat" shows "0 < Suc (n::nat)"
outside_def: fixes S :: "'b set" shows "outside S \<equiv> - S \<inter> {x. \<not> bounded (connected_component_set (- S) x)}"
assms_3: shows "x0 \<sqsubseteq> f x0"
assms_1: shows "agree_on G f\<^sub>1 f\<^sub>2"
I_intT_def: fixes \<sigma> :: "'tp"   and a :: "univ" shows "I_intT \<sigma> a \<equiv> infTp \<sigma> \<longrightarrow> Ik_intT \<sigma> a"
iterates_mono: fixes f :: "'b \<Rightarrow> 'a"   and F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'a" assumes "f \<in> ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F"   and "\<And>f. monotone (\<sqsubseteq>) (\<le>) f \<Longrightarrow> monotone (\<sqsubseteq>) (\<le>) (F f)" shows "monotone (\<sqsubseteq>) (\<le>) f"
ts_length_hint2: fixes x :: "nat \<Rightarrow> 'a list"   and i :: "nat" assumes "ts x" shows "length (x i) = Suc 0"
gt_diff: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wt t \<ge>\<^sub>p wt s"   and "head t >\<^sub>h\<^sub>d head s" shows "t >\<^sub>t s"
pbeta_Obj: fixes f' :: "Label \<Rightarrow> sterm option"   and f :: "Label \<Rightarrow> sterm option"   and L :: "char list set"   and T :: "Sigma.type" assumes "dom f' = dom f"   and "finite L"   and "\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (f' l) = \<sigma>[s,p] t)"   and "\<forall>l\<in>dom f. body (the (f l))" shows "Obj f T => Obj f' T"
simps_1: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and h :: "'b"   and a :: "'c" shows "heap_base.typeof_h typeof_addr h (Addr a) = map_option ty_of_htype (typeof_addr h a)"
real_of_ereal_enn2ereal: fixes x :: "ennreal" shows "real_of_ereal (enn2ereal x) = enn2real x"
valid_decomp_append: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "valid_decomp X ps"   and "valid_decomp X qs" shows "valid_decomp X (ps @ qs)"
Joints_reduce_aforms: fixes x :: "real list"   and X :: "(real \<times> real pdevs) list"   and prec :: "nat"   and t :: "(real \<times> real pdevs) list \<Rightarrow> nat \<Rightarrow> real list \<Rightarrow> bool" assumes "x \<in> Joints X" shows "x \<in> Joints (reduce_aforms prec t X)"
v11_vlrestriction: fixes A :: "V" shows "v11 (f \<restriction>\<^sup>l\<^sub>\<circ> A)"
Ints_of_int: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus z \<in> ring_1.Ints one plus zero minus"
IVT: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and y :: "'a"   and b :: "'b" assumes "f a \<le> y"   and "y \<le> f b"   and "a \<le> b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
step_z_norm_valid_dbm: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N> \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" shows "valid_dbm D'"
asymptotics4: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and x :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "x\<^sub>0 \<le> x" shows "1 \<le> (1 - ln x powr (- e / 2)) * 2"
abs_fract_def2: fixes q :: "'a fract" shows "\<bar>q\<bar> = (if q < 0 then - q else q)"
plus_bit0_def: fixes x :: "'a bit0"   and y :: "'a bit0" shows "x + y = Abs_bit0' (Rep_bit0 x + Rep_bit0 y)"
assms_6: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp" assumes "y = (scrut ? e1 : e2)" shows "P"
intK: fixes x :: "'s"   and f :: "'s option \<Rightarrow> real" shows "measure_pmf.expectation (K' (Some x)) f = measure_pmf.expectation (K x) (\<lambda>x. f (Some x))"
lhd_lfilter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lhd (lfilter P xs) = lhd (ldropWhile (Not \<circ> P) xs)"
prems: shows "abc_list_crsp (lm1 @ 0 \<up> Suc n) lm2"
Q_sets: fixes i :: "nat" shows "disjointed Q' i \<in> sets M"
equiv_class_self: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and a :: "'c" assumes "equiv A r"   and "a \<in> A" shows "a \<in> r `` {a}"
him: shows "h ` ({0..1} \<times> U) \<subseteq> S"
nn_integral_cong': fixes N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space N =simp=> f x = g x" shows "integral\<^sup>N N f = integral\<^sup>N N g"
distinct_5: fixes x2 :: "'b tree"   and x3 :: "'b tree" shows "Bal x2 \<noteq> Unbal x3"
int_pow_eq_id: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and i :: "int" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "(x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> i = \<one>\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub>) = (int (ord x) dvd i)"
nsub_RI_iconf_aux: fixes e :: "'a exp"   and sh :: "sheap"   and es :: "'a exp list" shows "(\<not> sub_RI e \<longrightarrow> (\<forall>e'. subexp_of e' e \<longrightarrow> \<not> sub_RI e' \<longrightarrow> iconf sh e') \<longrightarrow> iconf sh e) \<and> (\<not> sub_RIs es \<longrightarrow> (\<forall>e'. e' \<in> subexps es \<longrightarrow> \<not> sub_RI e' \<longrightarrow> iconf sh e') \<longrightarrow> iconfs sh es)"
has_subprob_density_measurable: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "f \<in> borel_measurable N"
init'_tableau_normalized: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init' init"   and "\<triangle> t" shows "\<triangle> (\<T> (init t))"
emeasure_bot: fixes X :: "'a set" shows "emeasure bot X = 0"
eq_onp_def: fixes R :: "'e \<Rightarrow> bool" shows "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
lappend_LNil_LNil: shows "<> $ <> = <>"
zero: assumes "vectorspace E" shows "(0::'a) \<in> V"
Class_widen_Class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list" shows "subtype P (Class C) (Class D) = P \<turnstile> C \<preceq>\<^sup>* D"
pat_isNone_1: fixes a :: "'b option"   and I :: "interface" shows "a = None \<equiv> (OP is_None :::\<^sub>i \<langle>I\<rangle>\<^sub>ii_option \<rightarrow>\<^sub>i i_bool) $ a"
trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_finite: fixes x :: "('c, 'd) stateful_strand_step" shows "finite (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p x)"
is_extensional: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and f :: "'a" assumes "functor A B F"   and "\<not> partial_magma.arr A f" shows "F f = partial_magma.null B"
abs_e1: shows "\<bar>e1\<bar> \<le> snd (trunc_bound_eucl p (fst (fst X) * fst (fst Y)))"
x: shows "x \<in>\<^sub>i interval_of (insertion e (map_poly real_of_float (tm_poly tm))) + real_interval (tm_bound tm)"
sep_conj_sep_true_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and h :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h" shows "sep_algebra.sep_conj plus sep_disj P (\<lambda>s. True) h"
length_rotate1: fixes xs :: "'a list" shows "length (rotate1 xs) = length xs"
assms_1: shows "B \<subseteq> U\<^sub>V\<^sub>S"
assms_1: fixes S :: "'a set" assumes "S \<in> \<A>" shows "path_connectedin X S"
bound_of_leI: fixes P :: "'c \<Rightarrow> real"   and c :: "real" assumes "\<And>x. P x \<le> c" shows "bound_of P \<le> c"
gfp_unfold: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "gfp f = f (gfp f)"
le_h: fixes x :: "real" assumes "0 \<le> x" shows "x powr (Re z - 1) * exp (- x) \<le> h x"
Re_power2: fixes x :: "complex" shows "Re (x\<^sup>2) = (Re x)\<^sup>2 - (Im x)\<^sup>2"
desargues_config_2D_rkRa_rkRb_rkRc_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and R :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {P, Q, R} = 2"   and "rk {Q, R} = 2"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {R, b} = 2"
valid_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<upsilon> X \<equiv> \<lambda>\<tau>. if X \<tau> = UML_Types.bot_class.bot \<tau> then false \<tau> else true \<tau>"
r_limr_recfn: shows "recfn 1 r_limr"
1: shows "frag_extend (frag_of \<circ> simplex_cone (p - Suc 0) v) (\<Sum>k = 0..p. frag_cmul ((- 1) ^ k) (frag_of (singular_face p k (oriented_simplex p l)))) = - (\<Sum>k = Suc 0..Suc p. frag_cmul ((- 1) ^ k) (frag_of (singular_face (Suc p) k (simplex_cone p v (oriented_simplex p l)))))"
FreeGroup_def: fixes S :: "'a set" shows "FreeGroup S \<equiv> {x. order.greater_eq S (fst ` set (freeword x))}"
arccos_minus: fixes x :: "real" assumes "- 1 \<le> x"   and "x \<le> 1" shows "arccos (- x) = pi - arccos x"