hcju/isabelleps huggingface dataset:
w: shows "\<forall>n\<in>insert_messages M \<union> {m'}. deps (Insert n) \<subseteq> I ` insert_messages M"
scaleC_conv_of_complex: fixes r :: "complex"   and x :: "'d" shows "r *\<^sub>C x = of_complex r * x"
arbitrage_processE: fixes Mkt :: "('a, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "arbitrage_process Mkt p" shows "\<exists>m. self_financing Mkt p \<and> trading_strategy p \<and> (\<forall>w\<in>space M. cls_val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < prob {w \<in> space M. 0 < cls_val_process Mkt p m w}"
TryCatch: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and e\<^sub>2 :: "char list exp"   and V :: "char list"   and e\<^sub>2' :: "char list exp"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, l\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> \<langle>e\<^sub>2,(h\<^sub>1, l\<^sub>1(V \<mapsto> Addr a))\<rangle> \<Rightarrow> \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2)\<rangle>" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2(V := l\<^sub>1 V))\<rangle>"
\<omega>_limit_set_in_compact_\<alpha>_limit_set_contained: fixes K :: "'a set"   and x :: "'a"   and z :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K"   and "z \<in> \<omega>_limit_set x" shows "\<alpha>_limit_set z \<subseteq> \<omega>_limit_set x"
has_field_decl_above: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "P \<turnstile> C has F:T (fm) in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
poly_DERIV: fixes p :: "'b poly"   and x :: "'b" shows "(poly p has_field_derivative poly (pderiv p) x) (at x)"
ordered_fmap_set_eq: fixes S :: "('a \<times> 'b) fset" assumes "is_fmap S" shows "fset_of_list (ordered_fmap S) = S"
continuous_bot: fixes f :: "'b \<Rightarrow> 'c" shows "continuous bot f"
assms_4: shows "is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"
weak_partial_order_bottom_axioms: shows "weak_partial_order_bottom (fpl L f)"
scaleC_cadjoint: fixes A :: "'a \<Rightarrow> 'b"   and a :: "complex" assumes "bounded_clinear A" shows "(\<lambda>t. a *\<^sub>C A t)\<^sup>\<dagger> = (\<lambda>s. cnj a *\<^sub>C (A\<^sup>\<dagger>) s)"
aux_lemma: fixes h :: "'a" assumes "h \<in> derived_set G H" shows "h = \<one>"
differentiable_inverse: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" assumes "f differentiable at x within s"   and "f x \<noteq> (0::'c)" shows "(\<lambda>x. inverse (f x)) differentiable at x within s"
pr_simps_4: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "arr (pr1 a b)"
len_empty_subview: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v' :: "view" shows "\<parallel>len v ts c\<parallel> = 0 \<and> v' \<le> v \<longrightarrow> \<parallel>len v' ts c\<parallel> = 0"
nonneg: fixes i :: "'b"   and n :: "'a" assumes "i \<in> I"   and "n \<in> A" shows "0 \<le> f n i"
simps: fixes T :: "ty" shows "the_Array (T\<lfloor>\<rceil>) = T"
scaleR_add_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "real"   and b :: "real"   and x :: "'d" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR (a + b) x = plus (scaleR a x) (scaleR b x)"
S_precise: fixes v :: "'n"   and M :: "'m set"   and P :: "'p multiset"   and k :: "nat" assumes "(v, M, P) \<in> S_cs fg k" shows "\<exists>p c' w. (([entry fg p], {#}), w, [v], c') \<in> trcl (trss fg) \<and> size P \<le> k \<and> {#[entry fg p]. p \<in># P#} \<subseteq># c' \<and> M = mon_w fg w"
intro: fixes \<I>_real :: "('y, 'aa) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and real_resource :: "('y + 'z, 'aa + 'ab) resource"   and \<I>_ideal :: "('ac, 'ad) \<I>"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and adv :: "real" assumes "\<I>_real \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res real_resource \<surd>"   and "\<I>_ideal \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res ideal_resource \<surd>"   and "\<I>_real, \<I>_ideal \<turnstile>\<^sub>C sim \<surd>"   and "pfinite_converter \<I>_real \<I>_ideal sim"   and "0 \<le> adv" shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"
assms: shows "accept\<^sub>G\<^sub>R (\<A> \<Sigma> \<phi>) w"
projecting_Always: fixes F :: "'a program"   and A :: "'a set" shows "projecting (\<lambda>G. reachable (extend h F \<squnion> G)) h F (Always (extend_set h A)) (Always A)"
iso_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b"   and "\<F>C.ide c" shows "\<F>C.iso (\<F>C.assoc a b c)"
finfun_All_All: fixes P :: "'b \<Rightarrow>f bool" shows "finfun_All P = All (($) P)"
reduce_system_sign_conditions_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "set (get_signs_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) = set (characterize_consistent_signs_at_roots p qs)"
invT_eq: fixes g :: "'g"   and h :: "'h" assumes "g \<in> G"   and "T g = h" shows "invT h = g"
simps_2: fixes sh :: "sheap"   and C :: "char list"   and e :: "'a exp" shows "iconf sh (Cast C e) = iconf sh e"
limXA: shows "limit_mat (\<lambda>n. X (n + k)) A m"
maximal_literal_is_unique_lemma: fixes t :: "'a trm"   and s :: "'a trm"   and C :: "'a literal set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and L1 :: "'a literal" assumes "\<not> equivalent_eq_exists t s C (int_clset S) \<sigma> L1" shows "maximal_literal_is_unique (t \<lhd> \<sigma>) (s \<lhd> \<sigma>) C L1 S \<sigma>"
fw_canonical: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and n :: "nat" assumes "cycle_free m n" shows "canonical (FW m n) n"
ce_rel_lm_2: fixes A :: "nat set" shows "ce_rel_to_set (ce_set_to_rel A) = A"
monad_fail_alt_def: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e"   and alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "monad_fail_alt return bind fail alt \<equiv> monad_fail return bind fail \<and> monad_alt return bind alt \<and> monad_fail_alt_axioms fail alt"
cases: fixes x :: "((int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray) \<times> nat" obtains f :: "int vec list \<times> int vec list" and mu :: "int iarray iarray" and d :: "int iarray" and i :: "nat" where "x = ((f, mu, d), i)"
span_zero: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale" shows "(0::'d) \<in> module.span scale S"
fun_dual2: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f = \<partial> \<circ> g) = (\<partial> \<circ> f = g)"
advantage_def: fixes \<A> :: "(unit, bool list + bool list \<times> 'hash, 'hash + unit) gpv" shows "UPF.advantage \<A> = spmf (UPF.game \<A>) True"
gcd_divides_r: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "somegcd G a b divides b"
domain_eq: shows "Domainp pcr_real = (\<lambda>x. realrel x x)"
sumI': fixes u :: "'c"   and U :: "'c set"   and v :: "'c"   and V :: "'c set" assumes "u \<in> U"   and "v \<in> V" shows "u + v \<in> U + V"
distinct_29: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> Stmt.If x51 x52 x53"
simps_2: fixes x :: "char list"   and e :: "aexp"   and P :: "'c" shows "post (x ::= e {P}) = P"
distinct_11: fixes x21 :: "POPLmark.type"   and x22 :: "trm"   and x41 :: "trm"   and x42 :: "trm" shows "(\<lambda>:x21. x22) \<noteq> x41 \<bullet> x42"
UP_smult_one: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)" shows "\<one>\<^bsub>R\<^esub> \<odot>\<^bsub>UP R\<^esub> p = p"
residue_primroot_mod: fixes n :: "nat"   and x :: "nat" shows "residue_primroot n (x mod n) = residue_primroot n x"
red_plus_keys_disjoint: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q"   and "keys p \<inter> keys r = {}" shows "red F (p + r) (q + r)"
close_subset: fixes C :: "('a, 'b) form set set" shows "C \<subseteq> close C"
bind_returns_result_E2: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and y :: "'c" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "pure f h" obtains x :: "'d" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>r y"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "f x < l"
homeomorphic_rel_frontiers_convex_bounded_sets: fixes S :: "'c set"   and T :: "'d set" assumes "convex S"   and "bounded S"   and "convex T"   and "bounded T"   and "aff_dim S = aff_dim T" shows "rel_frontier S homeomorphic rel_frontier T"
res_assoc2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<lhd> y \<cdot> z = x \<lhd> z \<lhd> y"
eq2: fixes r1 :: "'a"   and r2 :: "'a" assumes "r1 \<in> carrier R"   and "r2 \<in> carrier R" shows "f (r1 \<otimes>\<^bsub>R\<^esub> r2) = g (r1 \<otimes>\<^bsub>R\<^esub> r2)"
zero_le_power_abs: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times (abs a) n)"
Pi_cong_sets: fixes I :: "'c set"   and J :: "'c set"   and M :: "'c \<Rightarrow> 'd set"   and N :: "'c \<Rightarrow> 'd set" assumes "I = J"   and "\<And>x. x \<in> I \<Longrightarrow> M x = N x" shows "Pi I M = Pi J N"
major: fixes a :: "'a"   and b1 :: "'a"   and b2 :: "'a" assumes "a \<in> A"   and "a \<rightarrow>\<^sup>* b1"   and "a \<rightarrow>\<^sup>* b2"   and "is_final b1"   and "is_final b2" shows "b1 = b2"
app_unapp: 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 u :: "'b"   and u\<^sub>1 :: "'b"   and u\<^sub>2 :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "unapp u = Some (u\<^sub>1, u\<^sub>2)" shows "u = app u\<^sub>1 u\<^sub>2"
F1set3_bd_incr: fixes x :: "('e, 'f, 'd) F1" shows "|F1set3 x| \<le>o F1bd'"
alphaAbs_qAbs_iff_ex_distinct_qAFresh: fixes xs :: "'e"   and x :: "'f"   and X :: "('c, 'd, 'e, 'f, 'g) qTerm"   and xs' :: "'e"   and x' :: "'f"   and X' :: "('c, 'd, 'e, 'f, 'g) qTerm" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
rref_condition3: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. i < i + (1::'f) \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row (i + (1::'f)) A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ (i + (1::'f)) $ n \<noteq> (0::'d))"
inj_on_empty: fixes f :: "'d \<Rightarrow> 'e" shows "inj_on f {}"
prod_list_var_nonzero: fixes qs :: "real poly list" shows "prod_list_var qs \<noteq> 0"
prems: shows "s \<in> set (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A @ [a])))"
inext_Max: fixes I :: "nat set" assumes "finite I" shows "inext (Max I) I = Max I"
lsws': shows "length (snd (Strat ! x)) = length (take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) + 1"
mono: fixes c :: "'c"   and a :: "'c"   and d :: "'c"   and b :: "'c" assumes "c \<le> a"   and "d \<le> b" shows "max c d \<le> max a b"
finfun_comp_conv_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "('a, 'c) finfun" shows "finfun_comp g f = Abs_finfun (g \<circ> finfun_apply f)"
l_coset_carrier: fixes y :: "'a set"   and x :: "'a set"   and H :: "'a set set" assumes "y \<in> x <#\<^bsub>G Mod N\<^esub> H"   and "x \<in> carrier (G Mod N)"   and "subgroup H (G Mod N)" shows "y \<in> carrier (G Mod N)"
assms_1: shows "h1 = \<langle>hs1, x1, \<langle>\<rangle>\<rangle>"
last_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "xs \<noteq> []"   and "ys \<noteq> []"   and "length xs = length ys" shows "last (xs || ys) = (last xs, last ys)"
better_beta: fixes x :: "name"   and s :: "lam"   and t :: "lam" shows "(Lam [x].s) \<degree> t \<rightarrow>\<^sub>\<beta> s[x::=t]"
nhop_update_changed_kno_val: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and dsn :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt ip (dsn, kno, val, hops, nhip, {})" shows "the (nhop (update rt ip (dsn, kno, val, hops, nhip, {})) ip) = nhip"
deriv_stirling_integral_complex_of_real: fixes n :: "nat"   and x :: "real"   and j :: "nat" assumes "0 < n"   and "0 < x" shows "(deriv ^^ j) (stirling_integral n) (complex_of_real x) = complex_of_real ((deriv ^^ j) (stirling_integral n) x)"
bdd_lookup_subsetbdd: fixes N :: "bool list bdd list \<times> bool list"   and ws :: "bool list"   and q :: "bool list"   and a :: "nat" assumes "wf_nfa N (length ws)"   and "nfa_is_node N q"   and "a < length (fst N)" shows "bdd_lookup (subsetbdd (fst N) q (nfa_emptybdd (length q))) ws ! a = (\<exists>i<length q. q ! i \<and> bdd_lookup (fst N ! i) ws ! a)"
dim_ty_sem_Temperature_def: fixes uu :: "\<Theta> itself" shows "dim_ty_sem uu = \<^bold>\<Theta>"
simps_2: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list" shows "decomp_rm\<^sub>e\<^sub>s\<^sub>t (Decomp t # S) = decomp_rm\<^sub>e\<^sub>s\<^sub>t S"
inject: fixes a :: "'b"   and b :: "'d"   and a' :: "'b"   and b' :: "'d" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
not_zero_divisor_1: shows "\<not> zero_divisor (1::'b)"
wls_vsubst_ident: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "wls s X" shows "X #[z // z]_zs = X"
RESUMPTION_corec_def: fixes f :: "'j \<Rightarrow> ('h, 'g, 'f, 'i, 'j) resumption_corec"   and e :: "'j" shows "RESUMPTION_corec f e = (\<Union>n. RESUMPTION_corec_aux n f e)"
Max_Un: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "linorder.Max less_eq (A \<union> B) = ord.max less_eq (linorder.Max less_eq A) (linorder.Max less_eq B)"
Chn_vcomp: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>" shows "Chn (\<nu> \<bullet> \<mu>) = Chn \<nu> \<odot> Chn \<mu>"
e: fixes p :: "'b"   and q :: "'b" shows "((p, q) \<in> e) = (p = q)"
occs_elim: fixes v :: "'b"   and t :: "'b trm" assumes "\<not> Var v \<prec> t" shows "elim [(v, t)] v \<or> [(v, t)] \<doteq> []"
exp_def: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "exp b c = hf_to_ide (hexp (ide_to_hf b) (ide_to_hf c))"
dests_vD_inc_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:9, PRreq-:21, PRrep-:10} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> the (dests \<xi> ip) = inc (sqn (rt \<xi>) ip))) \<and> (l = PRerr-:1 \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> sqn (rt \<xi>) ip < the (dests \<xi> ip))))"
X: shows "limit (run\<^sub>t \<delta> q\<^sub>0 w) = f ` limit (run\<^sub>t \<delta>' q\<^sub>0' w)"
antisym: fixes a :: "'b list"   and b :: "'b list" assumes "a \<le>s b"   and "b \<le>s a" shows "a = b"
remdups_adj_Cons': fixes x :: "'g"   and xs :: "'g list" shows "remdups_adj (x # xs) = x # remdups_adj (dropWhile (\<lambda>y. y = x) xs)"
struct_maps_memI: fixes T :: "('a, 'b) partial_object_scheme"   and f :: "'a \<Rightarrow> 'c"   and S :: "('c, 'd) partial_object_scheme" assumes "\<And>x. x \<in> carrier T \<Longrightarrow> f x \<in> carrier S"   and "\<And>x. x \<notin> carrier T \<Longrightarrow> f x = undefined" shows "f \<in> struct_maps T S"
update_by_fun_raw_eq_update_by_raw: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and k :: "'b"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "oalist_raw.update_by_fun_raw rep_key_order k f xs = oalist_raw.update_by_raw rep_key_order (k, f (oalist_raw.lookup_raw rep_key_order xs k)) xs"
less_append: fixes s :: "'c list"   and t :: "'c list"   and a :: "'c list" assumes "s < t" shows "a @ s < a @ t"
k: fixes Q :: "nat \<Rightarrow> bool" assumes "Q \<in> M" shows "j1 < k Q \<and> Q (k Q)"
neg_less_divideC_eq: fixes c :: "complex"   and a :: "'b"   and b :: "'b" assumes "c < 0" shows "(a < b /\<^sub>C c) = (b < c *\<^sub>C a)"
dickson_gradingD_dgrad_set: fixes d :: "'c \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "almost_full_on (adds) (dgrad_set d m)"
analytic_on_const: fixes c :: "complex"   and S :: "complex set" shows "(\<lambda>z. c) analytic_on S"
cases: fixes a :: "'a set" assumes "finite a" obtains "ab__ = {}"   | A :: "'a set" and a :: "'a" where "ab__ = insert a A" and "finite A"
close_def: fixes C :: "'b fm set set" shows "close C \<equiv> {S. \<exists>S'\<in>C. S \<subseteq> S'}"
cinner_diff_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (minus y z) = cinner x y - cinner x z"
eta_reds_appL: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t" shows "s $ u \<rightarrow>\<^sub>\<eta>\<^sup>* t $ u"
assms_1: shows "WT_core \<I>_adv \<I>_usr I core"
A: shows "1 \<le> conv_abscissa fds_zeta"
sameDom_and_liftAll2_iff: fixes inp :: "'b \<Rightarrow> 'c option"   and inp' :: "'b \<Rightarrow> 'd option"   and phi :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(sameDom inp inp' \<and> liftAll2 phi inp inp') = (\<forall>i. inp i = None \<and> inp' i = None \<or> (\<exists>v v'. inp i = Some v \<and> inp' i = Some v' \<and> phi v v'))"
CHAOS_DF_refine_FD: fixes A :: "'c set" shows "CHAOS A \<sqsubseteq>\<^sub>F\<^sub>D DF A"
trancl_mono: fixes p :: "'c \<times> 'c"   and r :: "('c \<times> 'c) set"   and s :: "('c \<times> 'c) set" assumes "p \<in> r\<^sup>+"   and "r \<subseteq> s" shows "p \<in> s\<^sup>+"
M1: fixes p :: "'b"   and q :: "'b"   and s :: "'b"   and r :: "'b" assumes "p \<parallel> q"   and "p \<parallel> s"   and "r \<parallel> q" shows "r \<parallel> s"
closedin_imp_subset: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "closedin (subtopology U S) T" shows "T \<subseteq> S"
IH: shows "polynomial_of_mset xs - polynomial_of_mset (remove_powers xs) \<in> More_Modules.ideal polynomial_bool"
ide_def: fixes a :: "'a" shows "ide a \<equiv> a \<cdot> a \<noteq> null \<and> (\<forall>f. (f \<cdot> a \<noteq> null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> null \<longrightarrow> a \<cdot> f = f))"
top_on_inv_aval': fixes S :: "'av st option"   and X :: "char list set"   and e :: "aexp"   and a :: "'av" assumes "top_on\<^sub>o S X"   and "vars e \<subseteq> - X" shows "top_on\<^sub>o (inv_aval' e a S) X"
K_record_comp: fixes c :: "'e"   and f :: "'d \<Rightarrow> 'f" shows "(\<lambda>x. c) \<circ> f = (\<lambda>x. c)"
d_llp2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> d y \<cdot> x" shows "d x \<le> d y"
tObsC_tLast: fixes t :: "('a, 'es, 'as) BEState Trace" shows "tLast (tObsC t) = envObsC (es (tLast t))"
refine: fixes cmpk' :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res"   and cmpk :: "'c \<Rightarrow> 'c \<Rightarrow> comp_res"   and Rv :: "('c \<times> 'b) set"   and succi :: "'c \<Rightarrow> 'c list"   and E :: "('b \<times> 'b) set"   and vdi :: "'c"   and tgt :: "'b"   and v0i :: "'c"   and src :: "'b" assumes "eq_linorder cmpk'"   and "(cmpk, cmpk') \<in> Rv \<rightarrow> Rv \<rightarrow> comp_res_rel"   and "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"   and "(vdi, tgt) \<in> Rv"   and "(v0i, src) \<in> Rv" shows "(RETURN (dfs_impl3' cmpk succi v0i vdi), dfs E src tgt) \<in> \<langle>bool_rel\<rangle>nres_rel"
bottom_least: fixes L :: "('c, 'd) gorder_scheme" assumes "weak_partial_order_bottom L" shows "least L \<bottom>\<^bsub>L\<^esub> (carrier L)"
subspace_diff: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set"   and x :: "'d"   and y :: "'d" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
3: shows "t \<le> (r\<^sup>T * components ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g))\<^sup>T * (r\<^sup>T * components ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g))"