hcju/isabelleps huggingface dataset:
upd_invs_Some: fixes ts :: "'g \<Rightarrow> ('h \<times> 'i \<Rightarrow>f nat) option"   and tas :: "('g, 'j, 'k) new_thread_action list"   and I :: "'g \<Rightarrow> 'l option"   and t :: "'g"   and i :: "'l"   and x :: "'h \<times> 'i \<Rightarrow>f nat"   and Q :: "'l \<Rightarrow> 'g \<Rightarrow> 'j \<Rightarrow> 'k \<Rightarrow> bool" assumes "thread_oks ts tas"   and "I t = \<lfloor>i\<rfloor>"   and "ts t = \<lfloor>x\<rfloor>" shows "upd_invs I Q tas t = \<lfloor>i\<rfloor>"
pow_mono: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq b a"   and "less_eq zero b" shows "less_eq (power.power one times b n) (power.power one times a n) \<and> less_eq zero (power.power one times b n)"
intra_path_vp: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'" shows "n -as\<rightarrow>\<^sub>\<surd>* n'"
LIMSEQ_subseq_LIMSEQ: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c"   and f :: "nat \<Rightarrow> nat" assumes "X \<longlonglongrightarrow> L"   and "strict_mono f" shows "(X \<circ> f) \<longlonglongrightarrow> L"
p: shows "F, {#} \<Rightarrow> image_mset \<^bold>\<not> \<Gamma>' + \<Delta>'"
lift_Constrains: fixes i :: "nat"   and F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and A :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and B :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" shows "(lift i F \<in> lift_set i A Co lift_set i B) = (F \<in> A Co B)"
assms_3: shows "\<forall>c\<in>set cs. v c \<le> n"
emp_act: fixes p :: "'b" shows "\<alpha> (\<Squnion> {}) p = \<bottom>"
less_integer_code_2: fixes l :: "num" shows "(0 < Code_Numeral.Pos l) = True"
bad_times_def: shows "bad_times = (\<lambda>x. {n \<in> {Nf 1..}. \<exists>l\<in>{1..n}. u n x - u (n - l) x \<le> - (c0 * real l)} \<union> (\<Union>N\<in>{2..}. {n \<in> {Nf N..}. \<exists>l\<in>{Nf N..n}. u n x - u (n - l) x \<le> - (eps N * real l)}))"
add_match_split_fst': fixes m :: "'c match_expr"   and a :: "'c rule"   and rs :: "'c rule list" shows "add_match m (a # rs) = add_match m [a] @ add_match m rs"
unsat: shows "\<forall>F G. \<not> eval\<^sub>c\<^sub>s F G Cs"
below_refl: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.po below" shows "below x x"
standard_rules_edges: fixes lhs :: "('b Standard_Constant, nat) labeled_graph"   and rhs :: "('b Standard_Constant, nat) labeled_graph"   and C :: "'b set"   and L :: "'b Standard_Constant set"   and l :: "'b Standard_Constant"   and x :: "nat"   and y :: "nat" assumes "(lhs, rhs) \<in> standard_rules C L"   and "(l, x, y) \<in> edges rhs" obtains "l = S_Bot"   | "l = S_Top"   | "l = S_Idt"   | "l \<in> S_Const ` C"   | "l \<in> L"
eventually_ge_at_top: fixes c :: "'c" shows "eventually ((\<le>) c) at_top"
is_derivation_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_derivation \<NN> \<TT> \<RR> \<SS> u = CFG.derives \<NN> \<TT> \<RR> [\<SS>] u"
drlex_fun_strict_def: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "drlex_fun_strict s t = strict drlex_fun s t"
assms_4: shows "ptr |\<in>| object_ptr_kinds h"
ccTTree_mono: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls"   and G' :: "CoCalls" assumes "S \<subseteq> S'"   and "G \<sqsubseteq> G'" shows "ccTTree S G \<sqsubseteq> ccTTree S' G'"
onorm_scaleC_left: fixes r :: "'a \<Rightarrow> complex"   and f :: "'b" assumes "bounded_clinear r" shows "onorm (\<lambda>x. r x *\<^sub>C f) = onorm r * norm f"
ccBinds_eq: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "CoCallAnalysis.ccBinds ccExp \<Gamma>\<cdot>i = (\<Squnion>v\<mapsto>e\<in>map_of \<Gamma>. CoCallAnalysis.ccBind ccExp v e\<cdot>i)"
Lim_ident_at: fixes x :: "'a"   and s :: "'a set" assumes "at x within s \<noteq> bot" shows "netlimit (at x within s) = x"
spmf_restrict_spmf: fixes p :: "'h spmf"   and A :: "'h set"   and x :: "'h" shows "spmf (p \<upharpoonleft> A) x = (if x \<in> A then spmf p x else 0)"
3: shows "proots_count (map_poly complex_of_real (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:])) {x. 0 < Re x} = 0"
simps_1: fixes n :: "nat"   and n' :: "nat" shows "VNat n \<squnion> VNat n' = (if n = n' then Some (VNat n) else None)"
simps_4: fixes x :: "'a rexp"   and y :: "'a rexp" shows "lang (Plus x y) = lang x + lang y"
nth_mem: fixes n :: "nat"   and xs :: "'b list" assumes "n < \<^bold>|xs\<^bold>|" shows "xs ! n \<in> set xs"
induct: fixes fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and x1 :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and x2 :: "'form fset \<times> 'form"   and P :: "('form, 'rule, 'subst, 'var) itnode rose_tree \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool" assumes "iwf fc x1 x2"   and "\<And>n p s i \<Gamma> ants c. \<lbrakk>n \<in> sset nodes; Reg p |\<in>| outPorts n; list_all2 (\<lambda>ip t. iwf fc t ((\<lambda>h. subst s (freshen i (labelsOut n h))) |`| hyps_for n ip |\<union>| \<Gamma> \<turnstile> subst s (freshen i (labelsIn n ip))) \<and> P t ((\<lambda>h. subst s (freshen i (labelsOut n h))) |`| hyps_for n ip |\<union>| \<Gamma> \<turnstile> subst s (freshen i (labelsIn n ip)))) (inPorts' n) ants; fc n i s (\<Gamma> \<turnstile> c); c = subst s (freshen i p)\<rbrakk> \<Longrightarrow> P (INode n p i s ants) (\<Gamma> \<turnstile> c)"   and "\<And>c \<Gamma> s i. \<lbrakk>c |\<notin>| ass_forms; c |\<in>| \<Gamma>; c = subst s (freshen i anyP)\<rbrakk> \<Longrightarrow> P (HNode i s []) (\<Gamma> \<turnstile> c)" shows "P x1 x2"
WTrtCall: fixes 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> expr"   and Cs :: "char list list"   and es :: "expr list"   and Ts' :: "ty list" assumes "P,E,h \<turnstile> e : Class C"   and "P \<turnstile> C has least M = (Ts, T, m) via Cs"   and "P,E,h \<turnstile> es [:] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts" shows "P,E,h \<turnstile> Call e None M es : T"
red_rtrancl_mult: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "(punit.red F)\<^sup>*\<^sup>* p q" shows "(punit.red F)\<^sup>*\<^sup>* (local.punit.monom_mult c t p) (local.punit.monom_mult c t q)"
vec_of_poly_rev_shifted_def: fixes p :: "'b poly"   and n :: "nat"   and j :: "nat" shows "Resultant.vec_of_poly_rev_shifted p n j \<equiv> vec n (\<lambda>i. if i \<le> j \<and> j \<le> degree p + i then coeff p (degree p + i - j) else (0::'b))"
invar_def: fixes t :: "'t" shows "invar t = (inv t \<and> sorted (inorder t))"
invar_trie_impl_of: fixes t :: "('c, 'd) trie" shows "invar_trie (impl_of t)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_delete_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
simps_4: fixes V :: "'a set"   and s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list" shows "wf\<^sub>s\<^sub>t V (\<langle>assign: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S) = (fv t \<subseteq> V \<and> wf\<^sub>s\<^sub>t (V \<union> fv s) S)"
DiaI: fixes \<phi> :: "\<o>"   and v :: "i" assumes "\<exists>v. [\<phi> in v]" shows "[\<^bold>\<diamond>\<phi> in v]"
less_eq_option_Some_None: fixes x :: "'b" shows "(Some x \<le> None) = False"
rs_total: fixes s :: "nat \<Rightarrow> nat option"   and x :: "nat" assumes "total_cons s" shows "eval (total_cons.r_s s) [x] \<down>"
extremum_uniqueI: fixes a :: "'b list" assumes "a \<le>s \<epsilon>" shows "a = \<epsilon>"
obj_ty_CInst1: fixes obj :: "obj"   and C :: "qtname" assumes "tag obj = CInst C" shows "obj_ty obj = Class C"
int_of_uint32_code: fixes x :: "uint32" shows "int_of_uint32 x = int_of_integer (integer_of_uint32 x)"
gaugeD_2: fixes \<gamma> :: "'b \<Rightarrow> 'b set"   and x :: "'b" assumes "gauge \<gamma>" shows "open (\<gamma> x)"
monomial_decomp_sum_list_monomial_in_cone: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "monomial_decomp ps"   and "a \<in> sum_list ` listset (map cone ps)"   and "t \<in> keys a" obtains c :: "'b" and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" and U :: "'x set" where "(h, U) \<in> set ps" and "c \<noteq> (0::'b)" and "monomial c t \<in> cone (h, U)"
"1_4": shows "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
antisym: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b"   and "b \<le> a" shows "a = b"
select_convs_2: fixes tg_mut :: "mut"   and tg_t :: "t"   and more :: "'a" shows "tg_t \<lparr>tg_mut = tg_mut, tg_t = tg_t, \<dots> = more\<rparr> = tg_t"
SDTH: shows "Polynomial.smult p (sdiv_poly (S * D' + T * H' - 1) p) = S * D' + T * H' - 1"
le: shows "hypreal_of_real x + u \<le> hypreal_of_real y + v"
pseudominimal_ne_bottom: fixes P :: "'a set"   and x :: "'a" assumes "pseudominimal_in P x" shows "x \<noteq> bottom P"
vimageI2: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and A :: "'b set" assumes "f a \<in> A" shows "a \<in> f -` A"
p_pow_nonzero_1: fixes n :: "nat" shows "[p] \<cdot> \<one> [^] n \<in> carrier Zp"
assms_5: fixes q :: "'a list"   and r :: "'a list" assumes "q \<in> carrier (K [X])"   and "r \<in> carrier (K [X])"   and "p pdivides q \<otimes>\<^bsub>K [X]\<^esub> r" shows "p pdivides q \<or> p pdivides r"
gbinomial_0_left: fixes k :: "nat" shows "(0::'c) gchoose k = (if k = 0 then 1::'c else (0::'c))"
P2_yb_xf: shows "(P2, yb) = xf"
iso_in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "A.iso_in_hom f a b" obtains "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" and "A.iso f"
ereal_distrib: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<noteq> \<infinity> \<or> b \<noteq> - \<infinity>"   and "a \<noteq> - \<infinity> \<or> b \<noteq> \<infinity>"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "(a + b) * c = a * c + b * c"
ereal_mult_mono_strict': fixes a :: "ereal"   and c :: "ereal"   and b :: "ereal"   and d :: "ereal" assumes "0 < a"   and "0 < c"   and "a < b"   and "c < d" shows "a * c < b * d"
bottom_le: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c"   and B :: "'c"   and x :: "'c" assumes "lbv A r f T B" shows "B \<sqsubseteq>\<^bsub>r\<^esub> x"
catch_env_def: fixes catch :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and m1 :: "('c, 'm) envT"   and m2 :: "('c, 'm) envT" shows "local.catch_env catch m1 m2 = EnvT (\<lambda>r. catch (run_env m1 r) (run_env m2 r))"
asAbs_pick: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs set" assumes "goodAbs A" shows "asAbs (pick A) = A"
pr_int_mapk_used: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and b :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "IntMapK (S (Card n, n, run)) = Some b" shows "Pri_AgrK b \<in> U"
nres_rel_trans2: fixes a :: "'c nres"   and b :: "'c nres"   and i :: "'d nres"   and R :: "('d \<times> 'c) set" assumes "a \<le> b"   and "(i, a) \<in> \<langle>R\<rangle>nres_rel" shows "(i, b) \<in> \<langle>R\<rangle>nres_rel"
Right_Coset_unit: shows "H |\<cdot> \<one> = H"
elims: fixes x :: "'a list"   and y :: "cell list" assumes "tape_of_nat_list x = y" obtains "x = []" and "y = []"   | n :: "'a" where "x = [na__]" and "y = <na__>"   | n :: "'a" and v :: "'a" and va :: "'a list" where "x = n # v # va" and "y = <n> @ Bk # tape_of_nat_list (v # va)"
mat_gt_trans: fixes sd :: "nat"   and n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat" assumes "sd \<le> n"   and "mat_gt (\<succ>) sd A B"   and "mat_gt (\<succ>) sd B C"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "mat_gt (\<succ>) sd A C"
eta_case: fixes s :: "dB"   and dummy :: "dB"   and u :: "dB" assumes "\<not> free s 0"   and "s[dummy/0] => u" shows "\<exists>t'. Abs (s \<degree> Var 0) => t' \<and> t' \<rightarrow>\<^sub>\<eta>\<^sup>* u"
distinct_9: fixes ty1' :: "ty"   and ty2' :: "ty" shows "tBOOL \<noteq> ty1' \<rightarrow> ty2'"
add_implies_diff: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c + b = a" shows "c = a - b"
sum_lessThan_telescope: fixes f :: "nat \<Rightarrow> 'd"   and m :: "nat" shows "(\<Sum>n<m. f (Suc n) - f n) = f m - f 0"
poly_add_zero'_2: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_add [] p = local.normalize p"
f: shows "(\<lambda>t. h [fst t] \<otimes>\<^bsub>A\<^esub> h X [^]\<^bsub>A\<^esub> snd t) \<in> set (dense_repr p) \<rightarrow> carrier A"
sndI: fixes x :: "'e \<times> 'f"   and y :: "'e"   and z :: "'f" assumes "x = (y, z)" shows "snd x = z"
Card_order_empty: fixes r :: "('d \<times> 'd) set" assumes "Card_order r" shows "|{}| \<le>o r"
W_def: fixes j :: "nat" shows "W j = (if j < 2 then V j else V j - closure (V (j - 2)))"
1: shows "\<guillemotleft>local.map \<mu> : local.map (S.dom \<mu>) \<Rightarrow>\<^sub>B local.map (S.cod \<mu>)\<guillemotright>"
1: shows "length (T \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>) = length (S \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)"
STEPF_4: fixes r :: "('a, 'b) ta_rule" assumes "r \<in> \<delta>" shows "rcm' r = (if q \<in> set (rhsq r) then Some (the (rcm r) - 1) else rcm r)"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'b" assumes "F.A.ide a"   and "F.A.ide a'"   and "F.B.in_hom g (F a') (F a)" shows "\<exists>f. F.A.in_hom f a' a \<and> F f = g"
l_get_element_by_def: fixes get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_get_element_by get_element_by_id get_elements_by_tag_name to_tree_order \<equiv> (\<forall>h ptr iden element_ptr to. h \<turnstile> get_element_by_id ptr iden \<rightarrow>\<^sub>r Some element_ptr \<longrightarrow> h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to \<longrightarrow> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<in> set to) \<and> (\<forall>ptr tag h. pure (get_elements_by_tag_name ptr tag) h)"
dvd_const: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q"   and "q \<noteq> 0"   and "degree q = 0" shows "degree p = 0"
maxBlk_def: fixes s :: "state"   and p :: "Proc" shows "maxBlk s p = (SOME b. b \<in> nonInitBlks s p \<and> (\<forall>c\<in>nonInitBlks s p. bal c \<le> bal b))"
Succ_def: fixes Kl :: "'c list set"   and kl :: "'c list" shows "BNF_Greatest_Fixpoint.Succ Kl kl = {k. kl @ [k] \<in> Kl}"
not_less_Nil: fixes x :: "'b list" shows "\<not> x < []"
list_update_same_conv: fixes i :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "i < length xs" shows "(xs[i := x] = xs) = (xs ! i = x)"
invertible_mult_row: fixes q :: "'a"   and k :: "'a"   and a :: "'b" assumes "q * k = (1::'a)"   and "k * q = (1::'a)" shows "invertible (mult_row (mat (1::'a)) a q)"
inf_idem: fixes x :: "'d upred" shows "(x \<and> x) = x"
KerT: shows "KerT \<equiv> ker T \<inter> V"
ParCR: fixes c2 :: "('test, 'atom) com"   and s :: "'state"   and c2' :: "('test, 'atom) com"   and s' :: "'state"   and c1 :: "('test, 'atom) com" assumes "(c2, s) \<rightarrow>c (c2', s')" shows "(c1 | c2, s) \<rightarrow>c (c1 | c2', s')"
zs_def: shows "zs = map (\<lambda>p. if p \<in> {min i j<..max i j} then \<not> ys ! p else ys ! p) [0..<n]"
unx_some_lemmas_6: fixes u :: "'a"   and z :: "name" assumes "unfree u = Some z" shows "unapp u = None"
"1_1": shows "\<forall>a b. (P \<and>* P' \<and>* $ (n - m)) (a, b) \<longrightarrow> (F \<and>* Q) (a, b)"
steps_reflect_SN_on: fixes r :: "('b \<times> 'b) set"   and b :: "'b"   and a :: "'b" assumes "\<not> SN_on r {b}"   and "(a, b) \<in> r\<^sup>*" shows "\<not> SN_on r {a}"
N_d: fixes i :: "nat" assumes "i \<le> length fs" shows "d i \<le> N ^ i"
Aexp_subst_upd: fixes e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var"   and n :: "Arity" shows "\<A>\<^bsub>n\<^esub> e[y::=x] \<sqsubseteq> (\<A>\<^bsub>n\<^esub> e)(y := \<bottom>, x := up\<cdot>0)"
inner_right_distrib: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (plus y z) = inner x y + inner x z"
simps: fixes P :: "ast_problem"   and n :: "char list"   and args :: "object list" shows "ast_problem.resolve_instantiate P (PAction n args) = instantiate_action_schema (the (ast_domain.resolve_action_schema (domain P) n)) args"
meet_bot2: fixes x :: "'a" shows "x \<sqinter> \<bottom> = \<bottom>"
countable_iff_less_\<omega>1: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "countable (elts \<alpha>) = (\<alpha> < \<omega>1)"
to_tree_order_si_def: shows "to_tree_order_si \<equiv> l_to_tree_order_si\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order_si Shadow_DOM.get_child_nodes get_shadow_root"
sorted_a_hdw_n: shows "w \<noteq> [] \<longrightarrow> Sorted_Less.sorted (rev (a ! hd w))"
zero_closed: shows "\<zero>\<^bsub>V\<^esub> \<in> carrier V"
assms_8: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
two_less_eq_exp_length: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (numeral.numeral one plus (num.Bit0 num.One)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) LENGTH('c))"
ord_eq_le_trans: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" assumes "a = b"   and "b \<le>s c" shows "a \<le>s c"
axioms_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" assumes "functor A B F" shows "category A"
conj_subset_def: fixes A :: "'h set"   and P :: "'h \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> bool" shows "(A \<subseteq> {x. P x \<and> Q x}) = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
one_hop: fixes \<xi> :: "state"   and \<xi>' :: "state"   and ip :: "nat" assumes "\<forall>dip\<in>kD (rt \<xi>). the (nhop (rt \<xi>) dip) \<in> kD (rt \<xi>)"   and "\<xi>' = \<xi>\<lparr>rt := update (rt \<xi>) ip (0, unk, val, Suc 0, ip, {})\<rparr>" shows "\<forall>dip\<in>kD (rt \<xi>). the (nhop (update (rt \<xi>) ip (0, unk, val, Suc 0, ip, {})) dip) = ip \<or> the (nhop (update (rt \<xi>) ip (0, unk, val, Suc 0, ip, {})) dip) \<in> kD (rt \<xi>)"
responses_\<I>_mono: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('a, 'b) \<I>"   and x :: "'a" assumes "\<I> \<le> \<I>'"   and "x \<in> outs_\<I> \<I>" shows "responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x"
abs_vec_def: fixes x :: "('a, 'b) vec" shows "\<bar>x\<bar> = (\<chi>i. \<bar>x $ i\<bar>)"
new_element_no_child_nodes: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r []"
cxt_inv_stop: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and E :: "('a, 'c, 'd) Stmt list"   and c :: "('a, 'c, 'd) Stmt" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "sifum_lang.cxt_to_stmt E c = Stop" shows "c = Stop \<and> E = []"
f_def: fixes a :: "'a" shows "f a \<equiv> \<bar>dist x a - dist a y\<bar>"
a: shows "mix_pmf (local.my_U p) local.some_best local.some_worst \<succeq>[\<R>] mix_pmf (local.my_U q) local.some_best local.some_worst \<and> (mix_pmf (local.my_U q) local.some_best local.some_worst, mix_pmf (local.my_U p) local.some_best local.some_worst) \<notin> \<R>"
intro: fixes k :: "nat"   and program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and c :: "nat \<Rightarrow> ('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat" assumes "\<And>l. l < k \<Longrightarrow> computation.concurrent_step direct_memop_step empty_storebuffer_step program_step (\<lambda>p p' is sb. sb) (c (n + l)) (c (n + Suc l))" shows "program_trace program_step c n k"
div_add1_eq1_mod_0_right: fixes b :: "nat"   and m :: "nat"   and a :: "nat" assumes "b mod m = 0" shows "(a + b) div m = a div m + b div m"
to_mset_def: fixes t :: "('c, 'd) BinomialHeap" shows "to_mset t \<equiv> BinomialHeapStruc.queue_to_multiset (Rep_BinomialHeap t)"
Red_F_def: fixes N :: "('f \<times> 'l) set" shows "Red_F_\<G> N = (\<Inter>q\<in>Q. Red_F_\<G>_q q N)"