hcju/isabelleps huggingface dataset:
setdist_eq_0_sing_1: fixes x :: "'c"   and S :: "'c set" shows "(setdist {x} S = 0) = (S = {} \<or> x \<in> closure S)"
assms: shows "(b, c) \<in> A\<^sup>\<up>"
subpath_def: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.subpath g v1 es v2 subs \<equiv> sub_rel_of g subs \<and> v1 \<in> Graph.vertices g \<and> SubRel.ces v1 es v2 subs \<and> set es \<subseteq> edges g"
from_nat_0: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs 0 = zero"
not_zero_iB: shows "\<not> is_zero_row_upt_k i k B"
safe_sblit'_def: fixes src :: "'b array"   and si :: "integer"   and di :: "integer"   and len :: "integer" shows "safe_sblit' src si di len = safe_sblit src (nat_of_integer si) (nat_of_integer di) (nat_of_integer len)"
le_add2_hmset: fixes n :: "hmultiset"   and m :: "hmultiset" shows "n \<le> m + n"
bv_to_int_lower_limit_gt0: fixes w :: "bit list" assumes "0 < bv_to_int w" shows "2 ^ (length (norm_signed w) - 2) \<le> bv_to_int w"
PNT2_imp_PNT1_2: assumes "(\<lambda>x. \<pi> x * ln (\<pi> x)) \<sim>[at_top] (\<lambda>x. x)" shows "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)"
run_owner_iv_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "let (s', S, A, B, C, D) = run_owner_iv n s; SK = (Some S, {A, B}, {C, D}) in {Asset n} \<times> {Crypt (Auth_ShaKey n) (PriKey S), \<lbrace>Num 2, PubKey B\<rbrace>, \<lbrace>Num 4, PubKey D\<rbrace>} \<union> {Owner n} \<times> {\<lbrace>Num 1, PubKey A\<rbrace>, \<lbrace>Num 3, PubKey C\<rbrace>, SesKey SK, Crypt (SesK SK) (PubKey D)} \<subseteq> s' \<and> (Asset n, PubKey B) \<notin> s'"
b_def: shows "b = (\<lambda>n. 2 / sqrt pi * (if odd n then inverse (fact (n div 2)) else 0))"
diff_id: shows "diff k charts_eucl charts_eucl (\<lambda>x. x)"
false_imp: fixes H :: "'b pl set"   and p :: "'b pl"   and q :: "'b pl" assumes "H |- p -> false" shows "H |- p -> q"
dvd_div_eq_0_iff: fixes b :: "'a"   and a :: "'a" assumes "b dvd a" shows "(a / b = (0::'a)) = (a = (0::'a))"
exhaust: fixes y :: "BD" obtains "y = \<box>"   | "y = \<diamond>"
returns_result_select_result: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog" assumes "h \<turnstile> ok f" shows "h \<turnstile> f \<rightarrow>\<^sub>r |h \<turnstile> f|\<^sub>r"
vaflow: fixes F :: "real \<Rightarrow> variable \<Rightarrow> real"   and \<theta> :: "trm"   and \<zeta> :: "real" assumes "solves_ODE I F x \<theta>" shows "Uvariation (F \<zeta>) (F 0) {RVar x, DVar x}"
exhaust: fixes y :: "status" obtains "y = PreStart"   | "y = Running"   | "y = Finished"
lm084: fixes X :: "'a set"   and Y :: "'a set" assumes "finite X"   and "finite Y"   and "card (X \<inter> Y) = card X" shows "X \<subseteq> Y"
bind_option_cong: fixes x :: "'a option"   and y :: "'a option"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "x = y"   and "\<And>z. z \<in> set_option y \<Longrightarrow> f z = g z" shows "x \<bind> f = y \<bind> g"
deutsch_algo_def: shows "deutsch_algo \<equiv> (H \<Otimes> Quantum.Id 1) * (U\<^sub>f * (H * |Deutsch.zero\<rangle> \<Otimes> H * |Deutsch.one\<rangle>))"
dual_axiom_def: shows "dual_axiom \<equiv> \<langle> Game hgid1^d \<rangle> pusall \<leftrightarrow> ! (\<langle> Game hgid1 \<rangle> ! pusall)"
inj_on_def: fixes f :: "'i \<Rightarrow> 'j"   and A :: "'i set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
set_intros: fixes x2 :: "'a" shows "x2 \<in> set_option (Some x2)"
ss_rewrite: fixes h :: "'a"   and a :: "'a"   and e :: "nat"   and z1 :: "nat"   and z1' :: "nat"   and e' :: "nat"   and z2 :: "nat"   and z2' :: "nat" assumes "h \<in> carrier \<G>"   and "a \<in> carrier \<G>"   and "e < order \<G>"   and "\<^bold>g [^] z1 \<otimes> g' [^] z1' = a \<otimes> h [^] e"   and "e' < e"   and "\<^bold>g [^] z2 \<otimes> g' [^] z2' = a \<otimes> h [^] e'" shows "h = \<^bold>g [^] ((int z1 - int z2) * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>)) \<otimes> g' [^] ((int z1' - int z2') * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>))"
equivP_pick_preserves: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "P (pick X)"
rec_inseparableD: fixes A :: "nat set"   and B :: "nat set"   and X :: "nat set" assumes "rec_inseparable A B"   and "A \<subseteq> X"   and "B \<subseteq> - X"   and "computable X" shows "False"
is_OK_if_return_2: fixes b :: "bool"   and m :: "'a + 'b"   and x :: "'b" shows "isOK (if b then m else Inr x) = (\<not> b \<or> isOK m)"
coboundedI2: fixes b :: "'d"   and c :: "'d"   and a :: "'d" assumes "b \<le> c" shows "inf a b \<le> c"
SatP': fixes b :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>@ a p) at b in' set ((ps, a) # branch)"   and "A, n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
circuit_finite: fixes A :: "'a set" assumes "circuit A" shows "finite A"
euler_mascheroni_LIMSEQ_of_real: shows "(\<lambda>n. of_real (harm n - ln (real n))) \<longlonglongrightarrow> euler_mascheroni"
coregular_tcsigs_iff_exe_coregular_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss" shows "coregular_tcsigs (set cs) (translate_ars arss) = exe_coregular_tcsigs cs arss"
remdups_mset_singleton_sum: fixes a :: "'c"   and A :: "'c multiset" shows "remdups_mset (add_mset a A) = (if a \<in># A then remdups_mset A else add_mset a (remdups_mset A))"
suffix_append: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "xs \<le>s ys \<cdot> zs = (xs \<le>s zs \<or> (\<exists>xs'. xs = xs' \<cdot> zs \<and> xs' \<le>s ys))"
distr_PiM_component: fixes I :: "'a set"   and M :: "'a \<Rightarrow> 'b measure"   and i :: "'a" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"   and "i \<in> I" shows "distr (Pi\<^sub>M I M) (M i) (\<lambda>\<omega>. \<omega> i) = M i"
field_cont_on_gamma1_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (x, g2 x)} (\<lambda>a. F a \<bullet> i)"
poly_holomorphic_on: fixes p :: "complex poly"   and s :: "complex set" shows "poly p holomorphic_on s"
not_UNIV_le_Icc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and h :: "'b" assumes "class.no_top less_eq less" shows "\<not> UNIV \<subseteq> ord.atLeastAtMost less_eq l h"
graeffe_poly_impl_def: fixes f :: "'b poly" shows "graeffe_poly_impl f = graeffe_poly_impl_main ((- (1::'b)) ^ degree f) f"
zero_one_infty_not_equal_1: shows "1\<^sub>h \<noteq> \<infinity>\<^sub>h"
power2_less_eq_zero_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times a 2) zero = (a = zero)"
slim_simps_2: shows "parcs slim.slim = slim.slim_arcs"
R: shows "\<forall>x\<in>A. \<forall>y\<in>B. x \<otimes> y = y \<otimes> x"
L5: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" shows "map f (xs @ ys) = map f xs @ map f ys"
ODordTr1: fixes D :: "'c Order"   and E :: "'c Order" assumes "Worder D"   and "Worder E" shows "ordinal_number D \<sqsubset> ordinal_number E = (\<exists>b\<in>carrier E. ord_equiv D (Iod E (segment E b)))"
simps_2: fixes p :: "real \<times> real \<times> real"   and L :: "atomUni list" shows "separateAtoms (EqUni p # L) = (let (a, b, c, d) = separateAtoms L in (p # a, b, c, d))"
rel_compp: fixes R1 :: "'d \<Rightarrow> 'i \<Rightarrow> bool"   and S1 :: "'i \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'j \<Rightarrow> bool"   and S2 :: "'j \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_gpv (R1 OO S1) (R2 OO S2) = rel_gpv R1 R2 OO rel_gpv S1 S2"
small_big_trans': fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> l F g"   and "g \<in> L F h" shows "f \<in> L F h"
wf_trms_Var_range: fixes \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term" assumes "subst_range \<delta> \<subseteq> range Var" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
assms_5: shows "support_set pf \<subseteq> stocks Mkt"
sourceInRel: shows "\<forall>S1 S2. (S1, S2) \<in> SRel \<longrightarrow> (SourceTerm S1, SourceTerm S2) \<in> Rel"
abs_eq: fixes x :: "'d \<Rightarrow> 'e set" shows "outs_\<I> (Abs_\<I> x) = {out. x out \<noteq> {}}"
fmupd_idem: fixes a :: "'e"   and x :: "'f"   and y :: "'f"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "m(a \<mapsto>\<^sub>f y, a \<mapsto>\<^sub>f x) = m(a \<mapsto>\<^sub>f x)"
member_empty: shows "member RBT_Set2.empty = (\<lambda>_. False)"
eq_f: shows "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
red_preserves_hconf: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "P,E,h,sh \<turnstile> e : T"   and "P \<turnstile> h \<surd>" shows "P \<turnstile> h' \<surd>"
length_diff_Suc_exists: fixes xs :: "'b list"   and ys :: "'b list"   and m :: "nat" assumes "length xs - length ys = Suc m"   and "set ys \<subseteq> set xs"   and "distinct ys"   and "distinct xs" shows "\<exists>e. e \<in> set xs \<and> e \<notin> set ys"
free_eq_no_cars: shows "\<forall>ts v. (0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot)) = (0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>x. \<nexists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (clm ts) x = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1) \<and> True)))"
simple_extension_is_subring: fixes K :: "'a set"   and x :: "'a" assumes "subring K R"   and "x \<in> carrier R" shows "subring (simple_extension K x) R"
mkAutoSim_def: shows "mkAutoSim \<equiv> \<lambda>a. \<lparr>pInit = simInit a, pTrans = \<lambda>obs ec. SOME ec'. ec' \<in> set (simTrans a ec) \<and> simObs a ec' = obs, pAct = simAction a\<rparr>"
pos_le_divideC_eq: fixes c :: "complex"   and a :: "'b"   and b :: "'b" assumes "0 < c" shows "(a \<le> b /\<^sub>C c) = (c *\<^sub>C a \<le> b)"
elims: fixes x :: "nat"   and xa :: "'b list"   and y :: "nat" assumes "T_brauns x xa = y" obtains k :: "nat" and xs :: "'b list" where "x = k" and "xa = xs" and "y = (if xs = [] then 0 else let ys = take (2 ^ k) xs; zs = drop (2 ^ k) xs; ts = brauns (k + 1) zs in 4 * min (2 ^ k) (length xs) + T_brauns (k + 1) zs)"
simps_14: fixes v :: "MIR.num" shows "qfree (Eq v) = True"
left_def: shows "left = (\<lambda>S. \<Sum>x\<in>L S. jumpF h (at_left x))"
simps: fixes x :: "'b ext0"   and y :: "'b ext0" shows "x * y = x \<sqinter> y"
hyps_2: fixes B :: "('a, 'b, 'c) boundOutput"   and b :: "'d" assumes "ROut M C = ROut M B" shows "Prop b \<Psi> (P \<parallel> !P) M B"
bind_return_spmf: fixes x :: "'a spmf" shows "x \<bind> return_spmf = x"
simps_4: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and i :: "int" shows "P \<turnstile> typeof\<^bsub>h\<^esub> (Intg i) = \<lfloor>Integer\<rfloor>"
reflexive: fixes p :: "'c"   and A :: "('d, 'c) ngba" shows "p \<in> NGBA.reachable A p"
zero_one_enat_neq_1: shows "0 \<noteq> 1"
designI: assumes "\<And>b. b \<in># \<B> \<Longrightarrow> b \<noteq> {}"   and "\<B> \<noteq> {#}"   and "\<V> \<noteq> {}" shows "design \<V> \<B>"
hyp_edge_to_def: fixes c :: "'form"   and "is" :: "nat list" shows "hyp_edge_to c is = ((c, 0 # is), plain_ant anyP)"
InstanceOf_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and T :: "ty" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e instanceof T) h s s'"
exi_cong: fixes H :: "fmla set"   and A :: "fmla"   and A' :: "fmla"   and i :: "name" assumes "H \<turnstile> A IFF A'"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> exi i A IFF exi i A'"
not_dvd_pderiv: fixes p :: "'b poly" assumes "degree p \<noteq> 0" shows "\<not> p dvd pderiv p"
isnpolyh_unique: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "(\<forall>bs. Ipoly bs p = Ipoly bs q) = (p = q)"
relcompp_witness_1: assumes "((=) OO (=)) x x" shows "x = relcompp_witness (=) (=) (x, x)"
Cons_in_lex: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list"   and r :: "('b \<times> 'b) set" shows "((x # xs, y # ys) \<in> lex r) = ((x, y) \<in> r \<and> length xs = length ys \<or> x = y \<and> (xs, ys) \<in> lex r)"
estep_Mn: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and n :: "nat"   and f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Mn n f, xs, ls) # fs, rv)" shows "estep_Mn (encode_config c) = encode_config (step c)"
root_main'_upper: fixes n :: "int"   and x :: "int"   and y :: "int"   and b :: "bool" assumes "n \<le> x ^ p"   and "0 \<le> x"   and "0 \<le> n"   and "root_main' x n = (y, b)" shows "n < (y + 1) ^ p"
map_upt_Suc: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "map f [0..<Suc n] = f 0 # map (\<lambda>i. f (Suc i)) [0..<n]"
pathfinish_join: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" shows "pathfinish (g1 +++ g2) = pathfinish g2"
fail_def: fixes altc :: "'e cset \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd" shows "monad_altc_base.fail altc = altc cempty (\<lambda>_. undefined)"
prems_5: shows "older_seniors x n = older_seniors x (n + Suc m)"
PDG_path_cdep: fixes n :: "'node"   and n' :: "'node" assumes "PDG.PDG_edge sourcenode targetnode valid_edge Def Use standard_control_dependence n None n'" shows "n \<longrightarrow>\<^sub>d* n'"
ennreal_diff_le_mono_left: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a \<le> b" shows "a - c \<le> b"
monomial_power_map_scale: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and c :: "'d"   and n :: "nat" shows "monomial c t ^ n = monomial (c ^ n) (n \<cdot> t)"
that: assumes "\<sigma>' = Some (snd (Array.alloc (map f [0..<n]) (the_state \<sigma>)))"   and "r = fst (Array.alloc (map f [0..<n]) (the_state \<sigma>))"   and "Array.get (the_state \<sigma>') r = map f [0..<n]" shows "thesis"
items_def: fixes f :: "'c \<Rightarrow>\<^sub>0 'd" shows "items f = list.map (\<lambda>k. (k, lookup f k)) (sorted_list_of_set (keys f))"
bbox_def: fixes x :: "'a"   and y :: "'a" shows "antirange_semiring.bbox (\<lambda>x y. y \<cdot> x) ad x y = ad (x \<cdot> ad y)"
pre_oldest_configuration_initial_state: fixes n :: "nat"   and q :: "'b" assumes "Suc n \<in> pre_oldest_tokens q n" shows "q = q\<^sub>0"
bounded_WHILE_RES_rwof: fixes X :: "'a set" assumes "bounded_WHILE n cond step m0 = RES X"   and "x \<in> X" shows "rwof m0 cond step x"
assms_3: shows "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])"
find_lambda_measures_def: shows "find_lambda_measures \<equiv> measures [\<lambda>(l, c). mu - carrots_total c, \<lambda>(l, c). LEAST i. lambda \<le> c * 2 ^ i, \<lambda>(l, c). c - l]"
simps: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> a1 \<Rightarrow>\<^sub>z a2 = (a1 = [] \<and> a2 = Undecided \<or> (\<exists>m rs. a1 = Rule m Accept # rs \<and> a2 = Decision FinalAllow \<and> matches \<gamma> m p) \<or> (\<exists>m rs. a1 = Rule m Drop # rs \<and> a2 = Decision FinalDeny \<and> matches \<gamma> m p) \<or> (\<exists>m rs. a1 = Rule m Reject # rs \<and> a2 = Decision FinalDeny \<and> matches \<gamma> m p) \<or> (\<exists>m rs t. a1 = Rule m Log # rs \<and> a2 = t \<and> matches \<gamma> m p \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t) \<or> (\<exists>m rs t. a1 = Rule m Empty # rs \<and> a2 = t \<and> matches \<gamma> m p \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t) \<or> (\<exists>m rs t a. a1 = Rule m a # rs \<and> a2 = t \<and> \<not> matches \<gamma> m p \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t) \<or> (\<exists>m chain rs\<^sub>1 m' rs\<^sub>2 rrs t. a1 = Rule m (Call chain) # rrs \<and> a2 = t \<and> matches \<gamma> m p \<and> \<Gamma> chain = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2) \<and> matches \<gamma> m' p \<and> \<Gamma>,\<gamma>,p\<turnstile> rs\<^sub>1 \<Rightarrow>\<^sub>z Undecided \<and> \<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t) \<or> (\<exists>m chain rs X rrs. a1 = Rule m (Call chain) # rrs \<and> a2 = Decision X \<and> matches \<gamma> m p \<and> \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z Decision X) \<or> (\<exists>m chain rs rrs t. a1 = Rule m (Call chain) # rrs \<and> a2 = t \<and> matches \<gamma> m p \<and> \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z Undecided \<and> \<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t))"
axioms_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "span_in_category C (Cod \<mu>)"
simps_16: fixes V :: "'b"   and T :: "ty"   and vo :: "'d val option"   and e :: "('b, 'c, 'd) exp" shows "block_types {V:T=vo; e} = T # block_types e"
subset_trans: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "A \<subseteq> C"
closure_of_def: fixes X :: "'b topology"   and S :: "'b set" shows "X closure_of S \<equiv> {x \<in> topspace X. \<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y\<in>S. y \<in> T)}"
coherent_iff_coherent_Inv: fixes t :: "'c local.term" assumes "Can t" shows "coherent t = coherent (Inv t)"
blinfun_euclidean_eqI: fixes x :: "'d \<Rightarrow>\<^sub>L 'e"   and y :: "'d \<Rightarrow>\<^sub>L 'e" assumes "\<And>i. i \<in> Basis \<Longrightarrow> blinfun_apply x i = blinfun_apply y i" shows "x = y"
one_closed: shows "\<one>\<^bsub>relative_homology_group (int n) (nsphere n) (upper n)\<^esub> \<in> carrier (relative_homology_group (int n) (nsphere n) (upper n))"
cindexE_constI: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and c :: "real" assumes "\<And>t. \<lbrakk>a < t; t < b\<rbrakk> \<Longrightarrow> f t = c" shows "cindexE a b f = 0"
kE_satC: fixes c :: "('fsym, 'psym) lit list"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtC c"   and "I.wtE (kE \<xi>)" shows "I.satC (kE \<xi>) c = F.satC \<xi> c"
range_to_nat: shows "range Bij_Nat.to_nat = {0..<CARD('b)}"
conc_fst: fixes n :: "nat"   and w :: "'b list"   and x :: "nat \<Rightarrow> 'b" assumes "n < length w" shows "(w \<frown> x) n = w ! n"
tj_stack_pop_def: fixes tjs :: "'v list"   and u :: "'v" shows "tj_stack_pop tjs u = RETURN (tl (dropWhile ((\<noteq>) u) tjs), insert u (set (takeWhile ((\<noteq>) u) tjs)))"
uncross_positive_flip_def: fixes x :: "wall"   and y :: "wall" shows "uncross_positive_flip x y \<equiv> x = right_over \<and> y = left_over"
module_on_def: fixes S :: "'c set"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "module_on S scale \<equiv> ((\<forall>x y a. x \<in> S \<longrightarrow> y \<in> S \<longrightarrow> scale a (x + y) = scale a x + scale a y) \<and> (\<forall>x a b. x \<in> S \<longrightarrow> scale (a + b) x = scale a x + scale b x) \<and> (\<forall>x a b. x \<in> S \<longrightarrow> scale a (scale b x) = scale (a * b) x)) \<and> ((\<forall>x. x \<in> S \<longrightarrow> scale (1::'d) x = x) \<and> (\<forall>x y. x \<in> S \<longrightarrow> y \<in> S \<longrightarrow> x + y \<in> S)) \<and> (0::'c) \<in> S \<and> (\<forall>x a. x \<in> S \<longrightarrow> scale a x \<in> S)"
similar_mat_witD_4: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "A \<in> carrier_mat n n"
HierAuto_EmptySet: shows "(SOME x. True, {Abs_seqauto ({SOME x. True}, SOME x. True, {}, {})}, {}, [SOME x. True \<mapsto> {}]) \<in> {(D, F, E, G) |D F E G. HierAuto D F E G}"
assms_1: shows "SubstForm v \<lbrakk>\<guillemotleft>u\<guillemotright>\<rbrakk>e x x'"
induced_automorphism_C0: shows "\
 ` C0 = D0"
esqrt_mono2: fixes x :: "ereal"   and y :: "ereal" assumes "x \<le> y" shows "esqrt x \<le> esqrt y"
union: fixes A :: "'a set"   and B :: "'a set" shows "MaxR_opt (A \<union> B) = (case MaxR_opt A of None \<Rightarrow> MaxR_opt B | Some mA \<Rightarrow> Some (case MaxR_opt B of None \<Rightarrow> mA | Some mB \<Rightarrow> maxR mA mB))"
uint8_msb_test_bit: fixes x :: "uint8" shows "msb x = x !! 7"
ex_set_conv: fixes xs :: "'c list" shows "(\<exists>x. x \<in> set xs) = (xs \<noteq> [])"
ENR_convex_closed: fixes S :: "'b set" assumes "closed S"   and "convex S" shows "ENR S"
ptrm_prm_agreement_equiv: fixes \<pi> :: "'b prm"   and \<sigma> :: "'b prm"   and M :: "'b ptrm" assumes "\<And>a. a \<in> ds \<pi> \<sigma> \<Longrightarrow> a \<notin> ptrm_fvs M" shows "\<pi> \<bullet> M \<approx> \<sigma> \<bullet> M"
arr_char: fixes f :: "'b" shows "hhom\<^sub>D.arr f = \<guillemotleft>f : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a'\<guillemotright>"
simps_2: fixes v :: "'a list"   and va :: "'a list list" shows "interact (v # va) [] = concat (v # va)"
ln_div: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "ln (x / y) = ln x - ln y"
eq: fixes t :: "'a" shows "(\<Sum>v\<in>keys q. lookup q v when t + v = (0::'a)) = (lookup q (0::'a) when t = (0::'a))"
Cassoc: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "f \<approx>> g"   and "g \<approx>> h" shows "f ;; g ;; h = f ;; (g ;; h)"
leD: fixes y :: "'a list"   and x :: "'a list" assumes "border y x" shows "\<not> strict_border x y"