hcju/isabelleps huggingface dataset:
Star_1: fixes mr :: "mregex"   and ms :: "mregex" assumes "r = from_mregex mr \<phi>s"   and "safe_regex Past g (from_mregex mr \<phi>s)"   and "ms \<in> RPD mr" shows "safe_regex Past g (from_mregex ms \<phi>s)"
to_tree_order_is_l_to_tree_order: shows "l_to_tree_order Shadow_DOM.to_tree_order"
surj_f_inv_f: fixes f :: "'e \<Rightarrow> 'd"   and y :: "'d" assumes "surj f" shows "f (inv f y) = y"
strips_state_to_state_range_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c"   and a :: "'d"   and s' :: "'c \<times> 'd \<Rightarrow> bool option" assumes "is_valid_problem_sas_plus \<Psi>"   and "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "a \<in> \<R>\<^sub>+ \<Psi> v"   and "(v, a) \<in> dom s'"   and "\<forall>(v, a)\<in>dom s'. \<forall>(v, a')\<in>dom s'. s' (v, a) = Some True \<and> s' (v, a') = Some True \<longrightarrow> (v, a) = (v, a')" shows "((\<phi>\<^sub>S\<inverse> \<Psi> s') v = Some a) = the (s' (v, a))"
Abs_pos: fixes z :: "ant" shows "0 \<le> Abs z"
safe_rpd_fv_regex_le: fixes fv :: "'b \<Rightarrow> 'c set"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex"   and s :: "'b regex"   and test :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d" assumes "Regex.safe_regex fv safe Past Strict r"   and "s \<in> rpd test i r" shows "fv_regex fv r \<subseteq> fv_regex fv s"
commute: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> b = b \<oplus> a"
uii: shows "\<forall>k p q. upright (IncomingInfo (shd s) k p q)"
Quotient_alt_def2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (T x (Abs y) \<and> T y (Abs x))))"
map_of_zip_upd: fixes ys :: "'c list"   and xs :: "'d list"   and zs :: "'c list"   and x :: "'d"   and y :: "'c"   and z :: "'c" assumes "length ys = length xs"   and "length zs = length xs"   and "x \<notin> set xs"   and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" shows "map_of (zip xs ys) = map_of (zip xs zs)"
crename_fresh: fixes a :: "coname"   and M :: "trm"   and b :: "coname" assumes "a \<sharp> M" shows "M[a\<turnstile>c>b] = M"
assms: shows "\<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d)"
representant_exists: fixes A :: "('a set \<times> 'b) set" assumes "A \<in> rel.Partition" shows "\<exists>a\<in>Sigma I \<FF>. a \<in> A \<and> A = rel.Class a"
uconnectedV_trans: fixes E' :: "'a uprod set" shows "trans (Restr (uconnected E') verts)"
sublist_def: fixes xs :: "'c list"   and ys :: "'c list" shows "sublist xs ys = (\<exists>ps ss. ys = ps @ xs @ ss)"
case_eq_if: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and step :: "('g, 'f) step" shows "(case step of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = (if step = Done then f1 else if is_Skip step then f2 (un_Skip step) else f3 (un_Yield1 step) (un_Yield2 step))"
d_def: shows "d = dim_row (mk_poly (sylvester_mat p q))"
assoc_add_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "A + B + C = A + (B + C)"
efe: shows "real_of_int \<lfloor>Inum bs e\<rfloor> = Inum bs e"
norm_divide_numeral: fixes a :: "'b"   and w :: "num" shows "norm (a / numeral w) = norm a / numeral w"
after_summary_zmset_of_nonneg: fixes M :: "'t multiset"   and S :: "'sum antichain"   and t :: "'t" shows "0 \<le> zcount (after_summary (zmset_of M) S) t"
eq: shows "x \<^sup>@ a \<cdot> y \<^sup>@ b = z \<^sup>@ c"
finite_span: fixes S :: "'c set" assumes "finite S"   and "S \<subseteq> carrier M" shows "span S = {lincomb a S |a. a \<in> S \<rightarrow> carrier R}"
upper_closed: fixes y :: "'b" assumes "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "\<pi>\<^sub>*\<^bsub>G\<^esub> y \<in> carrier \<X>\<^bsub>G\<^esub>"
bisimSubstClosed: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "\<Psi> \<rhd> P \<sim>\<^sub>s Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<sim>\<^sub>s p \<bullet> Q"
meet_le_par: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> x \<parallel> y"
rtrancl_union_subset_rtrancl_union_trancl: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "(r \<union> s\<^sup>+)\<^sup>* = (r \<union> s)\<^sup>*"
longestImpWitnessPath_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c, 'd) configuration_scheme"   and loc1 :: "'a"   and loc2 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and t :: "'c" assumes "dataflow_topology summary results_in" shows "dataflow_topology.longestImpWitnessPath summary results_in c loc1 loc2 xs t = (dataflow_topology.impWitnessPath summary results_in c loc1 loc2 xs t \<and> (\<forall>loc' xs'. dataflow_topology.impWitnessPath summary results_in c loc' loc2 xs' t \<longrightarrow> length xs' \<le> length xs))"
sc_mem: fixes a :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "m \<in> carrier M" shows "a \<cdot>\<^sub>s m \<in> carrier M"
Abort_def: shows "Abort \<equiv> \<lambda>ab P s. if ab then 0 else 1"
netgmap_netgmap_not_rhs: fixes i :: "nat"   and s2 :: "'d net_state"   and sr :: "'d \<Rightarrow> 'e \<times> 'f"   and s1 :: "'d net_state" assumes "i \<notin> net_ips s2" shows "(fst (netgmap sr s1) ++ fst (netgmap sr s2)) i = fst (netgmap sr s1) i"
assms_2: shows "0 \<le> fls_subdegree g"
bounded_inner_imp_bdd_below: fixes s :: "'a set"   and a :: "'a" assumes "bounded s" shows "bdd_below ((\<lambda>x. x \<bullet> a) ` s)"
eval_eval_max: fixes s1 :: "state"   and e1 :: "expr"   and v1 :: "val"   and n1 :: "nat"   and t1 :: "state"   and s2 :: "state"   and e2 :: "expr"   and v2 :: "val"   and n2 :: "nat"   and t2 :: "state" assumes "s1 -e1\<succ>v1-n1\<rightarrow> t1"   and "s2 -e2\<succ>v2-n2\<rightarrow> t2" shows "s1 -e1\<succ>v1-max n1 n2\<rightarrow> t1 \<and> s2 -e2\<succ>v2-max n1 n2\<rightarrow> t2"
t_def: fixes seq :: "'observation list"   and obs :: "'observation" shows "t seq obs = length (filter ((=) obs) seq)"
set_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>set_rel)"
False: shows "\<not> (coprime a n \<and> coprime b n)"
assms_1: shows "i = None \<or> (\<exists>i'. i = Some i' \<and> i' \<in> fst ` set xs)"
common_subdiv_imp_gen_common_subdiv: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "common_boundary_sudivision_exists one_chain1 one_chain2" shows "common_sudiv_exists one_chain1 one_chain2"
splitAt_distinct_ab: fixes vs :: "'a list"   and a :: "'a list"   and b :: "'a list"   and ram :: "'a" assumes "distinct vs"   and "(a, b) = splitAt ram vs" shows "set a \<inter> set b = {}"
pair_construction_block_sizes: fixes B :: "'a set"   and I :: "'c set"   and f :: "'a set \<Rightarrow> ('a \<times> 'c) set multiset"   and \<K>' :: "int set"   and b :: "('a \<times> 'c) set" assumes "K_GDD (B \<times> I) (f B) \<K>' {{x} \<times> I |x. x \<in> B}"   and "B \<in># \<B>"   and "b \<in># f B" shows "int (card b) \<in> \<K>'"
update_simps_24: fixes T :: "(nat \<times> linear_poly) list"   and s :: "('a, 'b) state" shows "\<U>\<^sub>c (\<T>_update T s) = \<U>\<^sub>c s"
f4: shows "- 1 + 2 ^ LENGTH('b) = (- 1 + 2 ^ LENGTH('b)) mod 2 ^ LENGTH('b)"
starfun_Im: fixes f :: "'a \<Rightarrow> complex" shows "*f* (\<lambda>x. Im (f x)) = (\<lambda>x. hIm ((*f* f) x))"
nlistToTree_def: fixes xs :: "('b, 'c) Node list" shows "nlistToTree xs = lconsNlist xs Empty"
rep_eq: fixes x :: "('a, 'b) lifted_pair"   and xa :: "('a, 'b) lifted_pair" shows "(x \<le> xa) = (\<forall>f. triple.pairs_less_eq (Rep_lifted_pair x f) (Rep_lifted_pair xa f))"
a: shows "y\<^sup>\<star> \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star>"
mem_not_refl: fixes i :: "V" shows "i \<notin>\<^sub>\<circ> i"
next_def: fixes t :: "latin_type" shows "next t \<equiv> case t of Row \<Rightarrow> Col | Col \<Rightarrow> Num | Num \<Rightarrow> Row"
inner_octo_def: fixes x :: "octo"   and y :: "octo" shows "x \<bullet> y = Ree x * Ree y + Im1 x * Im1 y + Im2 x * Im2 y + Im3 x * Im3 y + Im4 x * Im4 y + Im5 x * Im5 y + Im6 x * Im6 y + Im7 x * Im7 y"
sum_diff1': fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "'b" assumes "finite {i \<in> I. f i \<noteq> (0::'c)}" shows "sum' f (I - {i}) = (if i \<in> I then sum' f I - f i else sum' f I)"
eq_inf_def: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" shows "w\<^sub>1 =\<^sub>I w\<^sub>2 \<equiv> w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2 \<and> w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1"
prod_diff1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "class.semidom_divide divide plus minus zero times one"   and "finite A"   and "f a \<noteq> zero" shows "comm_monoid_mult.prod times one f (A - {a}) = (if a \<in> A then divide (comm_monoid_mult.prod times one f A) (f a) else comm_monoid_mult.prod times one f A)"
wt_method_def: fixes P :: "'b prog"   and C :: "String.literal"   and Ts :: "ty list"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "'c instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "wt_method P C Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s \<equiv> 0 < length is \<and> length \<tau>s = length is \<and> check_types P mxs (1 + length Ts + mxl\<^sub>0) (map OK \<tau>s) \<and> wt_start P C Ts mxl\<^sub>0 \<tau>s \<and> (\<forall>pc<length is. P,T\<^sub>r,mxs,length is,xt \<turnstile> is ! pc,pc :: \<tau>s)"
szip_smap_left: fixes f :: "'e \<Rightarrow> 'c"   and xs :: "'e stream"   and ys :: "'d stream" shows "smap f xs ||| ys = smap (apfst f) (xs ||| ys)"
fds_abs_converges_integral: fixes f :: "'a fds"   and s :: "'a"   and c :: "'a" assumes "fds_abs_converges f s" shows "fds_abs_converges (fds_integral c f) s"
multiset: fixes empty :: "'c"   and is_empty :: "'c \<Rightarrow> bool"   and of_list :: "'d list \<Rightarrow> 'c"   and multiset :: "'c \<Rightarrow> 'd multiset"   and as_tree :: "'c \<Rightarrow> 'd Tree"   and remove_max :: "'c \<Rightarrow> 'd \<times> 'c"   and l :: "'c" assumes "Heap empty is_empty of_list multiset as_tree remove_max" shows "multiset l = Heap.multiset (as_tree l)"
compositor_simps_1: fixes r :: "'a"   and s :: "'a" assumes "ide r"   and "ide s"   and "src r = trg s" shows "Span.arr (CMP r s)"
Dube_is_GB_cofactor_bound_indets: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite F" shows "pp_pm.is_GB_cofactor_bound F (Dube (Suc (card (\<Union> (indets ` F)))) (maxdeg F))"
in_span_insert: fixes a :: "'b"   and b :: "'b"   and S :: "'b set" assumes "a \<in> cspan (insert b S)"   and "a \<notin> cspan S" shows "b \<in> cspan (insert a S)"
isoE: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" obtains g :: "'a \<times> 'a" where "B.VV.inverse_arrows f g"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "H.in_hom f a b" shows "H.arr f"
of_nat_zero_less_power_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less zero (power.power one times (semiring_1.of_nat one plus zero x) n) = (0 < x \<or> n = 0)"
sscan_scons_snth: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d stream"   and i :: "nat" shows "(a ## sscan f xs a) !! i = fold f (stake i xs) a"
negHorbit_Hom_indVfbasis_W: fixes T :: "'v \<Rightarrow> 'w" assumes "T ` V \<subseteq> W" shows "set (concat (negHorbit_homVfbasis T)) \<subseteq> W"
linear_on_diff_fun_push_forward: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real" assumes "linear_on src.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X" shows "linear_on dest.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) (push_forward X)"
lookup_monom_mult_zero: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'t" shows "lookup (monom_mult c (0::'a) p) t = c * lookup p t"
distinct_17: fixes x5 :: "char list" shows "Integer \<noteq> Class x5"
uniform_limit_theorem: fixes A :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and F :: "'b filter"   and l :: "'c \<Rightarrow> 'd" assumes "\<forall>\<^sub>F n in F. continuous_on A (f n)"   and "uniform_limit A f l F"   and "F \<noteq> bot" shows "continuous_on A l"
cobounded1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "semilattice_order f less_eq less" shows "less_eq (f a b) a"
main: fixes i :: "nat" assumes "i < nr" shows "row A i \<bullet> non_pivot_base A pp qj = (0::'a)"
exp_not_eq_zero: fixes x :: "'b" shows "exp x \<noteq> (0::'b)"
cis_times_cis_opposite: fixes \<phi> :: "real" shows "cis \<phi> * cis (- \<phi>) = 1"
\<phi>L_def: fixes \<phi> :: "'fmla" shows "\<phi>L \<phi> \<equiv> diag (imp P \<phi>)"
iwlsFSwSTR_imp_iwlsFSw: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" assumes "iwlsFSwSTR MOD" shows "iwlsFSw MOD"
dgrad_set_le_subset: fixes S :: "'c set"   and T :: "'c set"   and d :: "'c \<Rightarrow> nat" assumes "S \<subseteq> T" shows "dgrad_set_le d S T"
frestriction_def: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sub>\<bullet> A = set {[a, b]\<^sub>\<circ> |a b. a \<in>\<^sub>\<circ> A \<and> b \<in>\<^sub>\<circ> A \<and> [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
fset_inject: fixes x :: "'c fset"   and y :: "'c fset" shows "(fset x = fset y) = (x = y)"
star_subdist_var: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> + y\<^sup>\<star> \<le> (x + y)\<^sup>\<star>"
properfactor_divides: fixes a :: "'a"   and b :: "'a" assumes "properfactor (add_monoid R) a b" shows "a divides\<^bsub>add_monoid R\<^esub> b"
surjective: fixes r :: "('d, 'e, 'f, 'g, 'h) functor_scheme" shows "r = \<lparr>om = r \<^bsub>\<o>\<^esub>, am = r \<^bsub>\<a>\<^esub>, \<dots> = functor.more r\<rparr>"
eq: shows "rat_to_normalized_int_poly p = (lc, q)"
StaticCastRed: 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 e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "redp P E e s e' s'" shows "redp P E (\<lparr>C\<rparr>e) s (\<lparr>C\<rparr>e') s'"
one: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set" shows "\<one>\<^bsub>G\<^esub> \<in> generate G H"
Minf_le: fixes y :: "'b extended" shows "-\<infinity> \<le> y"
convex_bound_lt: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and a :: "'c"   and y :: "'c"   and u :: "'c"   and v :: "'c" assumes "class.linordered_semiring_1_strict plus zero minus less_eq less times one"   and "less x a"   and "less y a"   and "less_eq zero u"   and "less_eq zero v"   and "plus u v = one" shows "less (plus (times u x) (times v y)) a"
mat_index_conjugate: fixes i :: "nat"   and m :: "'a mat"   and j :: "nat" assumes "i < dim_row m"   and "j < dim_col m" shows "conjugate m $$ (i, j) = conjugate (m $$ (i, j))"
hasnext: fixes p :: "('s, 'm, 'p, 'l) seqp" assumes "p \<in> A" shows "\<exists>q. p \<leadsto>\<^bsub>\<Gamma>\<^esub> q \<and> q \<in> A"
Fix_lsl_iff: fixes x :: "'b" shows "(x \<in> Fix \<nu>) = (\<nu> x = x)"
push_down_older_seniors_senior: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "\<not> sink (token_run x n)"   and "\<not> sink (token_run y n)"   and "older_seniors x n = older_seniors y n" shows "senior x n = senior y n"
invar_rel_br: fixes b :: "'f \<Rightarrow> 'e set"   and a' :: "'d \<Rightarrow> 'f"   and I' :: "'d \<Rightarrow> bool"   and a :: "'c \<Rightarrow> 'e set"   and I :: "'c \<Rightarrow> bool" shows "\<langle>br a' I', br a I\<rangle>invar_rel b = br (\<lambda>(x, s). a x) (\<lambda>(x, s). I x \<and> I' s \<and> a x \<subseteq> b (a' s))"
disjnt_subset2: fixes X :: "'b set"   and Y :: "'b set"   and Z :: "'b set" assumes "disjnt X Y"   and "Z \<subseteq> Y" shows "disjnt X Z"
no_loops: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs H" shows "fst e \<noteq> snd e"
norm_lift_instr: fixes x :: "('dyn, 'c, 'fun, 'op, 'opinl) Inca.instr"   and \<Sigma>1 :: "'d option list"   and x' :: "('dyn, 'c, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and \<Sigma>2 :: "'d option list" assumes "lift_instr x \<Sigma>1 = Ok (x', \<Sigma>2)" shows "norm_instr x' = x"
P2_real_model_rewrite: fixes M :: "'a \<times> 'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "(bool \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a \<times> 'a) \<times> nat) \<times> 'g) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> 'e spmf)" shows "P2_real_model M \<sigma> z \<A> = P2_real_model' M \<sigma> z \<A>"
pochhammer_prod_rev: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a n = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero (n - i))) {1..n}"
m_c_def: fixes C :: "'a st option acom" shows "m\<^sub>c C = (\<Sum>a\<leftarrow>annos C. m\<^sub>o a (vars C))"
emeasure_finite: fixes M :: "'b measure"   and A :: "'b set" assumes "finite_measure M" shows "emeasure M A \<noteq> \<top>"
intros_1: fixes xs :: "'f llist" shows "lprefix LNil xs"
elims_2: fixes x :: "heap \<times> nat set" assumes "in_range x" obtains h :: "heap" and as :: "nat set" where "x = (h, as)" and "\<forall>a\<in>as. a < lim h"
dioph_dfs_mono: fixes ks :: "int list"   and l :: "int"   and z :: "nat option list \<times> int list"   and xs :: "int list"   and i :: "nat"   and k :: "nat"   and n :: "nat" assumes "dioph_invariant ks l z"   and "list_all (dioph_is_node ks l) xs"   and "fst z ! i = Some k" shows "fst (dioph_dfs.dfs n ks z xs) ! i = Some k"
outer_prod_le_one: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n"   and "inner_prod v v \<le> 1" shows "outer_prod v v \<le>\<^sub>L 1\<^sub>m n"
OrdP_Trans_lemma: fixes k :: "tm"   and i :: "tm"   and j :: "tm" shows "{OrdP k, i IN j, j IN k} \<turnstile> i IN k"
seqI': fixes f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>J c\<guillemotright>" shows "J.seq g f"
h2: shows "\<tt>\<tt> q' \<le> \<tt>\<tt> q"
list2FWpolicys_eq_sym: fixes Filter :: "((int \<times> int) set set, int) Combinators list" assumes "Filter \<noteq> []" shows "C (list2policyR (rev Filter)) = C (list2FWpolicy Filter)"
affine: fixes V :: "'a set" shows "affine V = (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
IH: shows "filter_mset P (mset_set A) = mset_set {x \<in> A. P x}"
box_axiom_def: shows "box_axiom \<equiv> [[Game hgid1]]pusall \<leftrightarrow> ! (\<langle> Game hgid1 \<rangle> ! pusall)"
IMP: fixes s :: "'a" assumes "I s"   and "\<not> b s"   and "(s, s\<^sub>0) \<in> R\<^sup>*" shows "\<Phi> s"
prems_3: shows "op\<^sub>1 \<noteq> op\<^sub>2"
vec_simps_16: fixes bi :: "'a"   and ai :: "'a"   and cg :: "'a" shows "bi * (ai * cg) = ai * (bi * cg)"
derivable_A: fixes x :: "'a set" assumes "Q.derivable x" shows "x \<in> Q"
rtc_implies_cbelow_on: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "dw_closed A"   and "a \<rightarrow>\<^sup>* b"   and "a \<in> A"   and "ord a c" shows "cbelow_on A ord c (\<leftrightarrow>) a b"
induct: fixes P :: "nat list \<Rightarrow> nat \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b) \<Rightarrow> 'b mat \<times> 'b mat \<times> 'b mat \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "nat"   and a2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a3 :: "'b mat \<times> 'b mat \<times> 'b mat" assumes "\<And>i bezout P A Q. P [] i bezout (P, A, Q)"   and "\<And>j xs i bezout P A Q. \<lbrakk>A $$ (i, i) dvd A $$ (j, j) \<Longrightarrow> P xs i bezout (P, A, Q); \<And>x xa y xb ya xc yb xd yc xe xf xg yd. \<lbrakk>\<not> A $$ (i, i) dvd A $$ (j, j); x = bezout (A $$ (i, i)) (A $$ (j, j)); (xa, y) = x; (xb, ya) = y; (xc, yb) = ya; (xd, yc) = yb; xe = diagonal_step_JNF A i j yc xd; xf = diagonal_step_PQ_JNF A i j bezout; (xg, yd) = xf\<rbrakk> \<Longrightarrow> P xs i bezout (xg * P, xe, Q * yd)\<rbrakk> \<Longrightarrow> P (j # xs) i bezout (P, A, Q)" shows "P a0 a1 a2 a3"
onlI: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'a) seqp"   and p :: "('c, 'd, 'b, 'a) seqp"   and P :: "'e \<times> 'a \<Rightarrow> bool"   and \<xi> :: "'e" assumes "\<And>l. l \<in> labels \<Gamma> p \<Longrightarrow> P (\<xi>, l)" shows "onl \<Gamma> P (\<xi>, p)"
sdrop_szip: fixes n :: "nat"   and s1 :: "'b stream"   and s2 :: "'c stream" shows "sdrop n (s1 ||| s2) = sdrop n s1 ||| sdrop n s2"
filterlim_at_bot: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "filterlim f at_bot F = (\<forall>Z. \<forall>\<^sub>F x in F. f x \<le> Z)"
call_return: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs\<^sub>1 :: "'c rule list"   and m' :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list" assumes "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> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> Undecided"
nf_inres_SPEC: fixes \<Phi> :: "'a \<Rightarrow> bool"   and x :: "'a" shows "nf_inres (SPEC \<Phi>) x = \<Phi> x"
assms: shows "atom ` domA \<Delta> \<sharp>* \<Gamma>"
Units_def: fixes G :: "('e, 'f) monoid_scheme" shows "Units G = {y \<in> carrier G. \<exists>x\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>}"
FR_def: shows "FR = R.lift_nnf_qe FR\<^sub>1"