hcju/isabelleps huggingface dataset:
comp_ipresIGWls: fixes h :: "'x \<Rightarrow> 'y"   and MOD :: "('z, 'aa, 'ab, 'ac, 'ad, 'ae, 'x, 'af, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'aj, 'ac, 'ak, 'al, 'y, 'am, 'an) model_scheme"   and h' :: "'y \<Rightarrow> 'ao"   and MOD'' :: "('ap, 'aq, 'ar, 'ac, 'as, 'at, 'ao, 'au, 'av) model_scheme" assumes "ipresIGWls h MOD MOD'"   and "ipresIGWls h' MOD' MOD''" shows "ipresIGWls (h' \<circ> h) MOD MOD''"
dom_eqI': fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "a \<in> VoV.domains f" shows "a = VoV.dom f"
r: shows "\<forall>x y. f (x \<cdot> - g y) \<le> f x \<cdot> - y \<and> g (y \<cdot> - f x) \<le> g y \<cdot> - x"
mset_\<sigma>: shows "mset (\<sigma> (f, Suc (length bef + length aft))) = mset lb + mset nlb"
assms_1: shows "x \<in> set (matrix_to_lpolies A)"
get_salt: fixes emBits :: "nat" assumes "176 + sLen \<le> emBits"   and "roundup emBits 8 * 8 - emBits \<le> 7" shows "generate_salt (maskedDB_zero (generate_DB (generate_PS emBits 160)) emBits) = salt"
C_def: shows "C = 1 - ln (ln 2) + I 2"
sums_def': fixes f :: "nat \<Rightarrow> 'b"   and s :: "'b" shows "f sums s = (\<lambda>n. sum f {0..n}) \<longlonglongrightarrow> s"
integrable_setdiff: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and S :: "'c set"   and j :: "'d"   and T :: "'c set" assumes "(f has_integral i) S"   and "(f has_integral j) T"   and "negligible (T - S)" shows "f integrable_on S - T"
vec_cmul: fixes c :: "'a"   and x :: "'a" shows "Finite_Cartesian_Product.vec (c * x) = c *s Finite_Cartesian_Product.vec x"
interval_eqI: fixes a :: "'b interval"   and b :: "'b interval" assumes "lower a = lower b"   and "upper a = upper b" shows "a = b"
left_edge_smooth: shows "(\<lambda>y. twoC (0, y)) piecewise_C1_differentiable_on {0..1}"
conv_radius_geI_ex: fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'a" assumes "\<And>r. \<lbrakk>0 < r; ereal r < R\<rbrakk> \<Longrightarrow> \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)" shows "R \<le> conv_radius f"
code: fixes sp :: "('b, 'a) sp\<^sub>\<nu>"   and s :: "'b stream" shows "run\<^sub>\<nu> sp s = (let ((h, sp), s) = run\<^sub>\<mu> (out sp) s in h ## run\<^sub>\<nu> sp s)"
subfield_m_inv_1: fixes K :: "'a list set"   and k :: "'a list" assumes "subfield K (K [X])"   and "k \<in> K - {\<zero>\<^bsub>K [X]\<^esub>}" shows "inv\<^bsub>K [X]\<^esub> k \<in> K - {\<zero>\<^bsub>K [X]\<^esub>}"
matT_vec_multI_def: fixes ze :: "'b"   and pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ti :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b list list"   and v :: "'b list" shows "matT_vec_multI ze pl ti m v = map (\<lambda>w. scalar_prodI ze pl ti w v) m"
atms_of_add_mset: fixes a :: "'a literal"   and A :: "'a literal multiset" shows "atms_of (add_mset a A) = insert (atm_of a) (atms_of A)"
interchange: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c"   and \<tau> :: "'c"   and \<sigma> :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (V \<nu> \<mu>)"   and "partial_magma.arr V (V \<tau> \<sigma>)" shows "H (V \<nu> \<mu>) (V \<tau> \<sigma>) = V (H \<nu> \<tau>) (H \<mu> \<sigma>)"
summable_dirichlet_prod: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "f 0 = (0::'a)"   and "g 0 = (0::'a)"   and "summable (\<lambda>n. norm (f n / nat_power n s))"   and "summable (\<lambda>n. norm (g n / nat_power n s))" shows "summable (\<lambda>n. norm (dirichlet_prod f g n / nat_power n s))"
mult_minus_right: fixes a :: "'a"   and b :: "'a" shows "a * - b = - (a * b)"
checker'_soundness_correctness: fixes a :: "real"   and al :: "float"   and au :: "float"   and b :: "real"   and bl :: "float"   and bu :: "float"   and c :: "real"   and cl :: "float"   and cu :: "float"   and d :: "real"   and dl :: "float"   and du :: "float"   and e :: "real"   and el :: "float"   and eu :: "float"   and f :: "real"   and fl :: "float"   and fu :: "float"   and p :: "nat" assumes "a \<in> {real_of_float al..real_of_float au}"   and "b \<in> {real_of_float bl..real_of_float bu}"   and "c \<in> {real_of_float cl..real_of_float cu}"   and "d \<in> {real_of_float dl..real_of_float du}"   and "e \<in> {real_of_float el..real_of_float eu}"   and "f \<in> {real_of_float fl..real_of_float fu}"   and "checker' p (Interval' al au) (Interval' bl bu) (Interval' cl cu) (Interval' dl du) (Interval' el eu) (Interval' fl fu)" shows "checker a b c d e f"
meet_connection: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "le x y = (dlat.meet le x y = x)"
st_data0_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and addr :: "word32" assumes "low_equal s1 s2" shows "st_data0 instr curr_win rd addr s1 = st_data0 instr curr_win rd addr s2"
lift_option_simps_1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "lift_option f (Some a) (Some b) = Some (f a b)"
bind_resumption_Pause: fixes out :: "'g"   and cont :: "'h \<Rightarrow> ('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.Pause out cont \<bind> f = Resumption.resumption.Pause out (\<lambda>inp. cont inp \<bind> f)"
order_lesseq_imp: fixes x :: "'c"   and y :: "'c" shows "(\<forall>z\<le>x. z \<le> y) = (x \<le> y)"
swapidseq_swap: fixes a :: "'a"   and b :: "'a" shows "swapidseq (if a = b then 0 else 1) (a \<rightleftharpoons>\<^sub>F b)"
assms_3: shows "invarNT (\<lambda>s. K s v)"
h4: assumes "Av < 0"   and "Bv \<le> 0" shows "(Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
fw_shortest: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "linordered_ab_monoid_add.cycle_free plus less_eq neutral m n"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "less_eq (linordered_ab_monoid_add.fw plus less_eq m n n n n i j) (plus (linordered_ab_monoid_add.fw plus less_eq m n n n n i k) (linordered_ab_monoid_add.fw plus less_eq m n n n n k j))"
assms: shows "mat_det (a, b, c, d) \<noteq> 0"
power_eq_imp_eq_base: fixes a :: "'a"   and n :: "nat"   and b :: "'a" assumes "a ^ n = b ^ n"   and "(0::'a) \<le> a"   and "(0::'a) \<le> b"   and "0 < n" shows "a = b"
initmissing_oreachable_netmask: fixes s :: "'s net_state"   and n :: "net_tree"   and U :: "(nat \<Rightarrow> 'g) \<Rightarrow> (nat \<Rightarrow> 'g) \<Rightarrow> bool" assumes "initmissing (netgmap sr s) \<in> oreachable (oclosed (opnet onp n)) (\<lambda>_ _ _. True) U"   and "net_ips s = net_tree_ips n" shows "netgmap sr s \<in> netmask (net_tree_ips n) ` oreachable (oclosed (opnet onp n)) (\<lambda>_ _ _. True) U"
G_sub: shows "extended_ord.punit.reduced_GB (homogenize None ` extend_indets ` F) \<subseteq> P[insert None (Some ` X)]"
in_subnet_def: fixes a :: "'a"   and S :: "'a set set" shows "a \<sqsubset> S = (\<exists>s\<in>S. a \<in> s)"
Suc_1: fixes S :: "'a set" assumes "card S = n"   and "\<And>x. x \<in> S \<Longrightarrow> [g x = a] (mod int p)" shows "[prod g S = a ^ n] (mod int p)"
lead_coeff_pCons_1: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "Polynomial.lead_coeff (pCons a p) = Polynomial.lead_coeff p"
infinite: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "infinite A" shows "F g A = \<^bold>1"
Abs_aezfun_inverse: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> aezfun_set" shows "aezfun (Abs_aezfun y) = y"
disc_7: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "isSend (Send x21 x22 x23 x24 x25 x26)"
prime_imp_coprime: fixes one :: "'d"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and gcd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and lcm :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and p :: "'d"   and n :: "'d" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "\<not> dvd.dvd times p n" shows "algebraic_semidom.coprime times one p n"
assms: fixes i :: "nat" shows "sigma_algebra (space M) (A i)"
arr_cod: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.arr C (partial_magma.cod C f)"
UN_lepoll_UN: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and C :: "'a \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<lesssim> C x"   and "pairwise (\<lambda>x y. disjnt (C x) (C y)) A" shows "\<Union> (B ` A) \<lesssim> \<Union> (C ` A)"
composable_if_connected: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "sources \<nu> \<inter> targets \<mu> \<noteq> {}" shows "\<nu> \<star> \<mu> \<noteq> null"
new_tv_le: fixes n :: "nat"   and m :: "nat"   and t :: "'b" assumes "n \<le> m"   and "new_tv n t" shows "new_tv m t"
PROTECT2_def: fixes x :: "'e"   and y :: "prop" shows "PROTECT2 x PROP y \<equiv> x"
induced_dec: shows "integral\<^sup>N M (induced_function A f) = (\<Sum>n. \<Sum>i\<le>n. d i n)"
right_neutral: fixes a :: "'d" shows "a * (1::'d) = a"
bot_2: shows "\<forall>q\<in>A. \<bottom> \<sqsubseteq> q"
oSuc_le_eq_less: fixes x :: "ordinal"   and y :: "ordinal" shows "(oSuc x \<le> y) = (x < y)"
unsat_imp_se_star_unsat: fixes c :: "('c, 'd) conf"   and ls :: "('c, 'd) label list"   and c' :: "('c, 'd) conf" assumes "se_star c ls c'"   and "\<not> Conf.sat c" shows "\<not> Conf.sat c'"
that: fixes y :: "real \<times> real" assumes "c.periodic_orbit y"   and "range (c.flow0 y) \<subseteq> trapC" shows "thesis"
diff_le_self_hmset: fixes m :: "hmultiset"   and n :: "hmultiset" shows "m - n \<le> m"
strip_guards_simps_8: fixes F :: "'f set"   and c :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "strip_guards F (DynCom c) = DynCom (\<lambda>s. strip_guards F (c s))"
circle_cube_boundary_explicit: shows "boundary circle_cube = {circle_left_edge, circle_right_edge, circle_bot_edge, circle_top_edge}"
add_eq_conv_diff: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a"   and N :: "'a multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> M = add_mset b (N - {#a#}) \<and> N = add_mset a (M - {#b#}))"
wf_design_implies: fixes \<B> :: "'b set multiset"   and V :: "'b set" assumes "\<And>b. b \<in># \<B> \<Longrightarrow> b \<subseteq> V"   and "\<And>b. b \<in># \<B> \<Longrightarrow> b \<noteq> {}"   and "finite V"   and "\<B> \<noteq> {#}"   and "V \<noteq> {}" shows "design V \<B>"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "nth_el (x # xs) 0 = x"
drop_bit_bin_cat_eq: fixes n :: "nat"   and w :: "int"   and v :: "int" shows "drop_bit n (concat_bit n w v) = v"
distinct_disc_2: fixes "term" :: "'b local.term" assumes "term = \<^bold>\<I>" shows "\<not> is_Prim term"
ovr_overshadow_right: fixes x :: "'s"   and y :: "'s"   and z :: "'s" shows "x \<triangleright> (y \<triangleright> z) = x \<triangleright> z"
G_2: shows "G' \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)"
obs: shows "\<parallel>A t\<parallel>\<^sub>o\<^sub>p \<le> Sup {\<parallel>A t\<parallel>\<^sub>o\<^sub>p |t. t \<in> T}"
uncountable_ball: fixes r :: "real"   and a :: "'b" assumes "0 < r" shows "uncountable (ball a r)"
map_of_filter_key_in: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "('b \<times> 'c) list" assumes "P x" shows "map_of (filter (\<lambda>(k, v). P k) xs) x = map_of xs x"
insert_remove: fixes A :: "'c set"   and x :: "'c" assumes "finite A" shows "card (insert x A) = Suc (card (A - {x}))"
le_inf_iff: fixes x :: "'b upred"   and y :: "'b upred"   and z :: "'b upred" shows "((y \<and> z) \<sqsubseteq> x) = (y \<sqsubseteq> x \<and> z \<sqsubseteq> x)"
included_def: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" shows "included V Q = (\<forall>(S, X)\<in>Q. S \<subseteq> V)"
2: fixes s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "s \<in> SMP (trms\<^sub>s\<^sub>s\<^sub>t S) - range Var"   and "t \<in> SMP (trms\<^sub>s\<^sub>s\<^sub>t S) - range Var"   and "\<exists>\<delta>. Unifier \<delta> s t" shows "\<Gamma> s = \<Gamma> t"
semiring_char_def: fixes uu :: "'a itself" shows "semiring_char uu = Gcd {n. of_nat n = (0::'a)}"
Aexp_edom: fixes e :: "exp"   and a :: "Arity" shows "edom (\<A>\<^bsub>a\<^esub> e) \<subseteq> fv e"
pell_valuation_neg_pos: fixes x :: "int"   and y :: "int" assumes "x < 0"   and "0 < y"   and "solution (x, y)" shows "pell_valuation (x, y) \<in> {- 1<..<0}"
delete_keys: fixes k :: "'a"   and al :: "('a \<times> 'b) list" shows "map fst (AList.delete k al) = removeAll k (map fst al)"
mset_set_set_mset': fixes M :: "'a multiset" assumes "\<forall>x. count M x \<le> 1" shows "mset_set (set_mset M) = M"
rel_mono: assumes "A \<le> A'"   and "B \<le> B'"   and "C \<le> C'" shows "rel_generat A B C \<le> rel_generat A' B' C'"
valid_path: shows "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"
is_postfixpoint_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_postfixpoint f x \<equiv> x \<le> f x"
src_src: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "src\<^sub>D (src\<^sub>D \<mu>) = src\<^sub>D \<mu>"
from_digits_digit: fixes x :: "nat"   and n :: "nat" assumes "x < base ^ n" shows "from_digits n (digit x) = x"
t_stop_def: shows "delayed_safe_distance.other.t_stop = - other.s' \<delta> / a\<^sub>o"
cod_comp: fixes g :: "'s"   and f :: "'s" assumes "seq g f" shows "cod (g \<cdot> f) = cod g"
distinct_6: fixes x41 :: "'b formula"   and x42 :: "'b formula"   and x1 :: "'b" shows "x41 \<^bold>\<and> x42 \<noteq> Atom x1"
components_red_subset: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" shows "component_of_term ` keys q \<subseteq> component_of_term ` keys p \<union> component_of_term ` Keys F"
AKcryptSKI: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, X\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "AKcryptSK authK servK evs"
lookup_plus_pp: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "lookup_pp (s + t) = lookup_pp s + lookup_pp t"
ordIso_reflexive: fixes r :: "('c \<times> 'c) set" assumes "Well_order r" shows "r =o r"
bisims1_map_Val_append: fixes P :: "'a prog"   and h :: "'heap"   and es' :: "'addr expr1 list"   and es'' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and es :: "'addr expr1 list"   and vs :: "'addr val list" assumes "P,es',h \<turnstile> (es'', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)"   and "length es = length vs" shows "P,es @ es',h \<turnstile> (map Val vs @ es'', xs) [\<leftrightarrow>] (stk @ rev vs, loc, length (compEs2 es) + pc, xcp)"
ts''_def_2: shows "\<C> \<turnstile> [Call_indirect j] : ts'' _> ts'"
language_def: shows "F.language \<equiv> {w. (!!) w \<Turnstile>\<^sub>p \<phi>}"
Quote_q_Eats: fixes y :: "hf"   and y' :: "hf"   and z :: "hf"   and z' :: "hf" assumes "Quote y y'"   and "Quote z z'" shows "Quote (y \<triangleleft> z) (q_Eats y' z')"
sign_changes_def: fixes ps :: "real poly list"   and x :: "real" shows "sign_changes ps x = length (remdups_adj (filter (\<lambda>x. x \<noteq> 0) (map (\<lambda>p. sgn (poly p x)) ps))) - 1"
measure_restrict_spmf: fixes p :: "'c spmf"   and A :: "'c set"   and X :: "'c set" shows "Sigma_Algebra.measure (measure_spmf (p \<upharpoonleft> A)) X = Sigma_Algebra.measure (measure_spmf p) (X \<inter> A)"
bigo_compose1: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and k :: "'c \<Rightarrow> 'a" assumes "f \<in> O(g)" shows "(\<lambda>x. f (k x)) \<in> O(\<lambda>x. g (k x))"
N_of_state_Liminf: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "N_of_state (Liminf_state Sts) = Liminf_llist (lmap N_of_state Sts)"
simps_3: fixes x :: "'a event"   and s :: "'a event list"   and X :: "'a event set" shows "setinterleaving (x # s, X, []) = (if x \<in> X then {} else {x # u |u. u setinterleaves ((s, []), X)})"
is_red_monomialE: fixes F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and v :: "'t"   and c :: "'e" assumes "is_red F (monomial c v)" obtains f :: "'t \<Rightarrow>\<^sub>0 'e" where "f \<in> F" and "f \<noteq> 0" and "lt f adds\<^sub>t v"
select_convs_2: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'a" shows "return_status \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = return_status"
proj_one: fixes t :: "cnf llist"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "\<exists>i. enat i < llength t \<and> active c (lnth t i)" shows "1 \<le> llength (dynamic_component.proj tCMP active c t)"
disc_58: fixes x8 :: "'a ltl" shows "is_LTLGlobal (G x8)"
eoutM_eout: fixes P :: "specID"   and M :: "chanID set"   and E :: "Expression" assumes "eoutM P M E" shows "eout P E"
H: shows "\<Delta> @ VarB V \<Colon> \<Gamma> \<turnstile> T <: U"
splice_Nil2: fixes xs :: "'a list" shows "splice xs [] = xs"
standard_lifting_def: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Inf_G :: "'b inference set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option" shows "standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I \<equiv> calculus Bot_G Inf_G entails_G Red_I_G Red_F_G \<and> standard_lifting_axioms Inf_F Bot_G Red_I_G Bot_F \<G>_F \<G>_I"
fmap_rel_def: fixes K :: "('e \<times> 'g) set"   and V :: "('f \<times> 'h) set" shows "\<langle>K, V\<rangle>fmap_rel \<equiv> {(m1, m2). (\<forall>i j. i |\<in>| fmdom m2 \<longrightarrow> (j, i) \<in> K \<longrightarrow> (the (fmlookup m1 j), the (fmlookup m2 i)) \<in> V) \<and> fset (fmdom m1) \<subseteq> Domain K \<and> fset (fmdom m2) \<subseteq> Range K \<and> (\<forall>i j. (i, j) \<in> K \<longrightarrow> (j |\<in>| fmdom m2) = (i |\<in>| fmdom m1))}"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   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_set_child_nodes type_wf set_child_nodes set_child_nodes_locs"   and "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "l_set_child_nodes_get_tag_name_axioms set_child_nodes_locs get_tag_name_locs" shows "l_set_child_nodes_get_tag_name type_wf set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs"
filter_filter: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nth_default_nth: fixes n :: "nat"   and xs :: "'b list"   and dflt :: "'b" assumes "n < length xs" shows "nth_default dflt xs n = xs ! n"
exists_exactly_one_following_state: fixes ev :: "('a, 'b, 'c) event"   and c :: "('a, 'b, 'c) configuration" assumes "can_occur ev c" shows "\<exists>!c'. c \<turnstile> ev \<mapsto> c'"
lift_option_simps_3: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a' :: "'b option" shows "lift_option f a' None = None"
subgroup_singular_relcycle: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "subgroup (singular_relcycle_set p X S) (chain_group p X)"
iterate_size_abort_def: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"   and n :: "nat" shows "iterate_size_abort it n = it (\<lambda>\<sigma>. \<sigma> < n) (\<lambda>x. Suc) 0"
prems_2: shows "ws t' \<noteq> \<lfloor>InWS w\<rfloor>"
pathstart_shiftpath: fixes a :: "real"   and g :: "real \<Rightarrow> 'a" assumes "a \<le> 1" shows "pathstart (shiftpath a g) = g a"
divide_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "x / y = x * inverse y"
dvd: shows "q ^ Suc i dvd (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
L_1: shows "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L"
agents'_def: shows "agents' = {A1, A2, A3, A4}"
cdf_lim_at_top: shows "(cdf M \<longlongrightarrow> prob (space M)) at_top"
set_eq: shows "set (map fst xs) = set (map fst ys)"
runit_agreement: fixes a :: "'a" assumes "ide a" shows "MC.runit a = \<r>[a]"
no_leaving: shows "(\<lambda>y. y + n + 1) ` R' `` (C \<inter> {..n}) \<subseteq> C"
interior_subset_union_intervals: fixes i :: "'a set"   and a :: "'a"   and b :: "'a"   and j :: "'a set"   and c :: "'a"   and d :: "'a"   and S :: "'a set" assumes "i = cbox a b"   and "j = cbox c d"   and "interior j \<noteq> {}"   and "i \<subseteq> j \<union> S"   and "interior i \<inter> interior j = {}" shows "interior i \<subseteq> interior S"
simps_5: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_WbisT (c1 | c2) = (SC_WbisT c1 \<and> SC_WbisT c2 \<or> noWhile (c1 | c2) \<and> SC_discr (c1 | c2) \<or> SC_siso (c1 | c2))"
rtranclp_mono: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "r \<le> s" shows "r\<^sup>*\<^sup>* \<le> s\<^sup>*\<^sup>*"
cmaps_false: fixes \<theta>_c :: "(coname \<times> name \<times> trm) list"   and c :: "coname"   and x :: "name"   and P :: "trm" assumes "\<theta>_c cmaps c to Some (x, P)"   and "c \<sharp> \<theta>_c" shows "False"
sorted_list_of_set_3: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
semiring_normalization_rules_5: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "plus zero a = a"
comm_monoid_axioms: shows "Groups.comm_monoid (+) (0::'a)"
a_consistent_app\<^sub>2: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, Lam [y]. e, Arg x # S)" shows "a_consistent (ae, pred\<cdot>a, as) (\<Gamma>, e[y::=x], S)"
that: fixes d :: "real"   and B :: "real" assumes "0 < d"   and "0 < B"   and "\<And>x y. \<lbrakk>x \<in> {a--b}; dist x y \<le> d\<rbrakk> \<Longrightarrow> B \<le> f y \<bullet> rot (b - a)" shows "thesis"