hcju/isabelleps huggingface dataset:
cases: fixes a :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) \<times> (nat \<times> ('sf, 'sc, 'sz) step) list" assumes "proof_ok a" obtains D :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" and S :: "(nat \<times> ('sf, 'sc, 'sz) step) list" where "a = (D, S)" and "deriv_ok (start_proof D) S"
sdom_dom: fixes i :: "nat"   and j :: "nat" assumes "strict_dom i j" shows "local.dom i j"
eventually_at_top_not_equal: fixes c :: "'a" shows "\<forall>\<^sub>F x in at_top. x \<noteq> c"
allDefs_def: fixes g :: "'g"   and n :: "'node" shows "allDefs g n \<equiv> defs g n \<union> phiDefs g n"
rep_eq: fixes x :: "nat"   and xa :: "uint32" shows "Rep_uint32 (drop_bit x xa) = drop_bit x (Rep_uint32 xa)"
atleast_free_Suc: fixes m :: "'c \<Rightarrow> 'd option"   and n :: "nat" shows "atleast_free m (Suc n) = (\<exists>a. m a = None \<and> (\<forall>b. atleast_free (m(a \<mapsto> b)) n))"
no_rewritten_fermat3: fixes x :: "int" shows "\<nexists>v w. v ^ 3 + w ^ 3 = x ^ 3 \<and> v * w * x \<noteq> 0 \<and> even x \<and> coprime v w"
id_transfer: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun A A id id"
density_context_if_dens: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and M :: "(nat \<Rightarrow> val) \<Rightarrow> val measure"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and b :: "bool" assumes "density_context V V' \<Gamma> \<delta>"   and "has_parametrized_subprob_density (state_measure (V \<union> V') \<Gamma>) M (count_space (range BoolVal)) f" shows "density_context V V' \<Gamma> (if_dens \<delta> f b)"
Ln_exp: fixes z :: "complex" assumes "- pi < Im z"   and "Im z \<le> pi" shows "ln (exp z) = z"
hyps: shows "P \<turnstile> C \<preceq>\<^sup>* D"
LBINT_I0i_1_div_plus_square: shows "LBINT x=0..\<infinity>. 1 / (1 + x\<^sup>2) = pi / 2"
adj_sur_inv: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "top_pres ((`) f) = (f \<circ> g = id)"
shows_list_arctic_delta_def: fixes ps :: "'a arctic_delta list" shows "shows_list ps = showsp_list shows_prec 0 ps"
Un_Diff: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<union> B - C = A - C \<union> (B - C)"
to_fun_Fun_mult: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "to_fun (x \<otimes>\<^bsub>P\<^esub> y) = to_fun x \<otimes>\<^bsub>function_ring (carrier R) R\<^esub> to_fun y"
assms: fixes i :: "'b" assumes "i \<in> I" shows "integrable M (f i)"
DBM_val_bounded_def: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat" shows "u \<turnstile>\<^bsub>v,n\<^esub> m \<equiv> Le (0::'d) \<preceq> m 0 0 \<and> (\<forall>c. v c \<le> n \<longrightarrow> dbm_entry_val u None (Some c) (m 0 (v c)) \<and> dbm_entry_val u (Some c) None (m (v c) 0)) \<and> (\<forall>c1 c2. v c1 \<le> n \<and> v c2 \<le> n \<longrightarrow> dbm_entry_val u (Some c1) (Some c2) (m (v c1) (v c2)))"
a: shows "n x \<cdot> t y \<sqsubseteq> n z"
basis_reduction_add_rows_bound: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list"   and fs' :: "int vec list" assumes "LLL_bound_invariant True upw i fs"   and "basis_reduction_add_rows upw i fs = fs'"   and "i < m" shows "LLL_bound_invariant True False i fs'"
simps_2: fixes n :: "nat"   and s :: "'a stream" shows "sdrop (Suc n) s = sdrop n (stl s)"
coboundedI1: fixes c :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "c \<subseteq># a" shows "c \<subseteq># a \<union># b"
join_def': fixes G :: "graph"   and p :: "nat list"   and q :: "nat list" assumes "joinable G p q" shows "path_join G p q = butlast p @ q"
TER: shows "\<E> (TER g) = \<E> (TER f)"
ide_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "dual_category C" shows "partial_magma.ide (dual_category.comp C) a = partial_magma.ide C a"
partition_on_remove_pt: fixes A :: "'b set"   and G :: "'b set set"   and x :: "'b" assumes "partition_on A G" shows "partition_on (A - {x}) {g - {x} |g. g \<in> G \<and> g \<noteq> {x}}"
filterlim_at_top_gt: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" shows "filterlim f at_top F = (\<forall>Z>c. \<forall>\<^sub>F x in F. Z \<le> f x)"
bridge_compl_cases: fixes v :: "nat"   and v' :: "nat"   and xs :: "(nat \<times> nat) set"   and k :: "'c"   and j :: "'c"   and p :: "('b, 'c) fm"   and w :: "nat"   and w' :: "nat" shows "bridge k j xs v v' p = (\<^bold>\<diamond> Nom j) \<and> bridge k j xs w w' (\<^bold>\<not> p) = (\<^bold>\<not> (\<^bold>\<diamond> Nom k)) \<or> (\<exists>r. bridge k j xs v v' p = (\<^bold>\<not> (\<^bold>@ j r)) \<and> bridge k j xs w w' (\<^bold>\<not> p) = (\<^bold>\<not> \<^bold>\<not> (\<^bold>@ k r))) \<or> (\<exists>r. bridge k j xs v v' p = (\<^bold>@ k r) \<and> bridge k j xs w w' (\<^bold>\<not> p) = (\<^bold>\<not> (\<^bold>@ j r))) \<or> bridge k j xs v v' p = p \<and> bridge k j xs w w' (\<^bold>\<not> p) = (\<^bold>\<not> p)"
associated_polynomials_iff: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "p \<sim>\<^bsub>K [X]\<^esub> q = (\<exists>k\<in>K - {\<zero>}. p = [k] \<otimes>\<^bsub>K [X]\<^esub> q)"
greatest_mem: fixes L :: "('c, 'd) gorder_scheme"   and l :: "'c"   and A :: "'c set" assumes "greatest L l A" shows "l \<in> A"
rep_eq: fixes x :: "'a permutation" shows "(\<rightarrow>) (- x) = the_inv ((\<rightarrow>) x)"
assms: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (\<circle>(\<lambda>s. P s \<or> Q s)) s)) \<sigma>"
t_wise_balance_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int" shows "t_wise_balance point_set block_collection grouping index \<equiv> proper_design point_set block_collection \<and> t_wise_balance_axioms point_set block_collection grouping index"
Suc_3: fixes x :: "'a" assumes "x \<in> S" shows "[g x = a] (mod int p)"
lres_mult_sub_lres_lres: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x / (z * y) \<le> x / y / z"
lex_fun_refl: fixes s :: "'a \<Rightarrow> 'c" shows "lex_fun s s"
fls_nth_compose_power: fixes d :: "nat"   and f :: "'b fls"   and n :: "int" assumes "0 < d" shows "fls_nth (fls_compose_power f d) n = (if int d dvd n then fls_nth f (n div int d) else (0::'b))"
pred_fun_def: fixes A :: "'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool" shows "pred_fun A B = (\<lambda>f. \<forall>x. A x \<longrightarrow> B (f x))"
le_less_trans: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<preceq>\<^bsub>D\<^esub> b"   and "b \<prec>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
vector_2_var: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "is_vector y" shows "(x \<cdot> y\<^sup>T) ; z = (x \<cdot> y\<^sup>T) ; (y \<cdot> z)"
not_less_iff_gr_or_eq: fixes x :: "'a"   and y :: "'a" shows "(\<not> x < y) = (y < x \<or> x = y)"
power_le_imp_le_exp: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "less_eq (power.power one times a m) (power.power one times a n)" shows "m \<le> n"
A1: fixes p2 :: "nat \<times> nat" assumes "(p2, p1) \<in> lex_p" shows "c_assoc_have_key (pr_gr (loc_upb (fst p2) (snd p2))) (c_pair (fst p2) (snd p2)) = 0"
IF: fixes \<sigma> :: "'c" assumes "I {} \<sigma>" shows "P \<sigma>"
c_def: fixes xs :: "'b list"   and fs :: "'b list"   and i :: "nat" shows "??.Newton_Interpolation.c xs fs i = ??.Newton_Interpolation.xij_f xs fs 0 i"
relax_mono: fixes u :: "'v"   and D :: "'v \<Rightarrow> enat"   and v :: "'v" shows "relax_outgoing u D v \<le> D v"
scalar_prod_right_unit: fixes i :: "nat"   and n :: "nat"   and v :: "'a vec" assumes "i < n" shows "v \<bullet> unit_vec n i = v $ i"
intensional_simps_12: fixes P :: "'b \<Rightarrow> bool" shows "P \<longrightarrow> #True \<equiv> #True"
finestpart_def: fixes X :: "'a set" shows "finestpart X = (\<lambda>x. {x}) ` X"
second_summand_disjointness: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CEM"   and "\<not> O z (x \<oplus> y)" shows "\<not> O z y"
valid_SE_def: fixes \<sigma> :: "'d"   and m :: "'d \<Rightarrow> (bool \<times> 'd) option" shows "(\<sigma> \<Turnstile> m) = (m \<sigma> \<noteq> None \<and> fst (the (m \<sigma>)))"
toplevel_summands_PLUS: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd rexp" assumes "xs \<noteq> []" shows "toplevel_summands (PLUS (map f xs)) = (\<Union>r\<in>set xs. toplevel_summands (f r))"
AE_iff_null_sets: fixes N :: "'a set"   and M :: "'a measure" assumes "N \<in> sets M" shows "(N \<in> null_sets M) = (AE x in M. x \<notin> N)"
rbt_lookup_rbtreeify: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and kvs :: "('c \<times> 'd) list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map fst kvs)"   and "distinct (map fst kvs)" shows "ord.rbt_lookup less (rbtreeify kvs) = map_of kvs"
dom_in_domains: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C"   and "partial_magma.domains C f \<noteq> {}" shows "partial_magma.dom C f \<in> partial_magma.domains C f"
act_simp: fixes I :: "'a \<Rightarrow> bool"   and s :: "'b"   and s' :: "'b"   and a :: "'a" shows "act I s s' a = I a"
right_inverse_eq: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "(a / b = (1::'a)) = (a = b)"
shift1_isnpoly: fixes p :: "'b poly" assumes "isnpoly p"   and "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p)"
viette2_monic: fixes b :: "real"   and c :: "real"   and \<xi>1 :: "real"   and \<xi>2 :: "real" assumes "0 \<le> b\<^sup>2 - 4 * c"   and "\<xi>1\<^sup>2 + b * \<xi>1 + c = 0"   and "\<xi>2\<^sup>2 + b * \<xi>2 + c = 0"   and "\<xi>1 \<noteq> \<xi>2" shows "\<xi>1 * \<xi>2 = c"
h_def: shows "h = (\<lambda>t. g (t + a))"
beta_star_preserves_term_ok: fixes r :: "Core.term"   and s :: "Core.term"   and thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s"   and "term_ok thy r" shows "term_ok thy s"
ax_iu: fixes h :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" shows "\<box> h \<and> (A \<and>* B) h \<longrightarrow> A h"
has_vector_derivative_cnj: fixes f :: "real \<Rightarrow> complex"   and f' :: "complex"   and z :: "real"   and A :: "real set" assumes "(f has_vector_derivative f') (at z within A)" shows "((\<lambda>z. cnj (f z)) has_vector_derivative cnj f') (at z within A)"
init_satisfies_tableau: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "\<langle>\<V> (init t)\<rangle> \<Turnstile>\<^sub>t t"
incl_def: fixes f :: "'s" shows "incl f = (arr f \<and> Dom f \<subseteq> Cod f \<and> f = incl_of (local.dom f) (cod f))"
of_real_0: shows "of_real 0 = (0::'a)"
var_lookup_in: fixes x :: "'a \<Longrightarrow> 'b"   and A :: "'b"   and A' :: "'c" shows "get\<^bsub>$x\<^esub> (A, A') = get\<^bsub>x\<^esub> A"
axioms_1: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "AGM_FullContraction Cn contraction true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "AGM_Contraction Cn contraction"
alw_state_eq_smap: fixes s :: "nat option"   and ss :: "state stream" shows "alw (state_eq s) ss = alw (\<lambda>ss. shd ss = s) (smap statename ss)"
toplevel_summands_nonempty: fixes r :: "'b rexp" shows "toplevel_summands r \<noteq> {}"
catch_optionT_def: shows "catch = catch_option Monad_Overloading.return Monad_Overloading.bind"
eq0: fixes n :: "nat" assumes "i - m \<le> n" shows "(\<Prod>k\<le>n. f (k + m)) = (0::'a)"
qring_zero: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "\<zero>\<^bsub>R /\<^sub>r I\<^esub> = I"
one_mod_card_int: shows "1 mod int CARD('b) = 1"
add_diff_cancel: fixes a :: "'a"   and b :: "'a" shows "a + b - b = a"
assms_1: shows "s \<in> oreachable A PS PU"
take_map: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "take n (map f xs) = map f (take n xs)"
Ord_ord_of_nat: fixes k :: "nat" shows "Ord (ord_of_nat k)"
1: fixes z :: "'a" shows "z \<in> Px \<longrightarrow> {y \<in> Px. y \<sqsubset> z} = {y \<in> P. y \<sqsubset> z}"
hyps_1: shows "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>"
obtain_s_as: shows "x \<in> (\<lambda>(s, as). Inf (PLS s as)) ` {(s, as). s \<in> valid_states PROB \<and> as \<in> valid_plans PROB}"
subterm_inj_on_alt_def': fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subterm_inj_on \<theta> (subst_domain \<theta>) = (inj_on \<theta> (subst_domain \<theta>) \<and> (\<forall>s\<in>subst_range \<theta>. \<forall>u\<in>subst_range \<theta>. (\<exists>v. v \<sqsubseteq> s \<and> v \<sqsubseteq> u) \<longrightarrow> s = u))"
higher_eq_zero_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(higher p v = 0) = (\<forall>u. v \<prec>\<^sub>t u \<longrightarrow> lookup p u = (0::'c))"
simps: fixes a :: "16 word"   and b :: "16 word"   and c :: "16 word"   and d :: "16 word"   and e :: "16 word"   and f :: "16 word"   and g :: "16 word"   and h :: "16 word" shows "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (let lss = goup_by_zeros [a, b, c, d, e, f, g, h]; max_zero_seq = foldr (\<lambda>xs. max (length xs)) lss 0 in Let (if 1 < max_zero_seq then List_replace1 (replicate max_zero_seq 0) [] lss else lss) List_explode)"
SUP_real_ennreal: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "A \<noteq> {}"   and "bdd_above (f ` A)" shows "(\<Squnion>a\<in>A. ennreal (f a)) = ennreal (\<Squnion> (f ` A))"
dropWhile_map: fixes P :: "'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
set_remdups_sorted: fixes xs :: "'a list" assumes "sorted xs" shows "set (remdups_sorted xs) = set xs"
inserted_item_ident: fixes a :: "'a"   and xs :: "'a list"   and e :: "'a"   and i :: "'a option" assumes "a \<in> set (insert_spec xs (e, i))"   and "a \<notin> set xs" shows "a = e"
sep_split_substate: fixes \<sigma>1 :: "'a"   and \<sigma>2 :: "'a"   and \<sigma> :: "'a" assumes "\<sigma>1,\<sigma>2\<triangleright>\<sigma>" shows "\<sigma>1 \<preceq> \<sigma> \<and> \<sigma>2 \<preceq> \<sigma>"
sorted_list_len_lt: fixes x :: "'a set"   and y :: "'a set" assumes "x \<subset> y"   and "finite y" shows "length (sorted_list_of_set x) < length (sorted_list_of_set y)"
we_1_def: shows "we_1 = nat_to_ce_set e_1"
converse_rtrancl_on_into_rtrancl_on: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c"   and F :: "'c set" assumes "(a, b) \<in> r"   and "(b, c) \<in> rtrancl_on F r"   and "a \<in> F" shows "(a, c) \<in> rtrancl_on F r"
cont_nec_fact2_1: fixes v :: "i" shows "[WeaklyContingent O! in v]"
exhaust: fixes y :: "'e \<times> 'f" obtains a :: "'e" and b :: "'f" where "y = (a, b)"
assms_4: shows "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host"
is_span: shows "C.span f.cod.leg0 f.cod.leg1"
invertibleE: fixes u :: "'a \<Rightarrow> 'a"   and thesis :: "bool" assumes "symmetric.sub.invertible u"   and "\<And>v. \<lbrakk>compose S u v = identity S \<and> compose S v u = identity S; v \<in> G\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> G" shows "thesis"
closed_cdi_Inc: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "nat \<Rightarrow> 'c set" assumes "closed_cdi \<Omega> M"   and "range A \<subseteq> M"   and "A 0 = {}"   and "\<And>n. A n \<subseteq> A (Suc n)" shows "\<Union> (range A) \<in> M"
widen_subcls: fixes C :: "char list"   and D :: "char list"   and P :: "'a prog" assumes "P \<turnstile> C \<preceq>\<^sup>* D" shows "P \<turnstile> Class C \<le> Class D"
eventually_nhds_uniformity: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "eventually P (nhds x) = (\<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> P y)"
VARtoCorrectSYSTEM_def: shows "VARtoCorrectSYSTEM \<equiv> \<forall>v. \<forall>Z\<in>AbstrLevel level0 \<union> AbstrLevel level1. v \<in> VAR Z \<longrightarrow> (\<forall>x\<in>VARto v. x \<in> OUT Z)"
ide_mkIde_finite: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set" assumes "A \<subseteq> Univ"   and "finite A" shows "ide (mkIde A)"
reindex_bij_witness_not_neutral: fixes S' :: "'b set"   and T' :: "'c set"   and S :: "'b set"   and i :: "'c \<Rightarrow> 'b"   and j :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g :: "'b \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "finite S'"   and "finite T'"   and "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"   and "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"   and "\<And>a. a \<in> S' \<Longrightarrow> g a = (0::'d)"   and "\<And>b. b \<in> T' \<Longrightarrow> h b = (0::'d)"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "sum g S = sum h T"
induced_induce: fixes vs :: "'a set" assumes "vs \<subseteq> verts (G \<restriction> S)" shows "induced_subgraph ((G \<restriction> S) \<restriction> vs) (G \<restriction> S)"
simps_18: fixes f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> 'i \<Rightarrow> 'f"   and f3 :: "'j \<Rightarrow> ('k \<Rightarrow> 'i) \<Rightarrow> 'f"   and f4 :: "('g, 'h, 'j, 'k) resumption \<Rightarrow> 'f"   and x21 :: "'h"   and x22 :: "'i" shows "(case Linear_corec x21 x22 of Terminal_corec x \<Rightarrow> f1 x | Linear_corec x xa \<Rightarrow> f2 x xa | Branch_corec x xa \<Rightarrow> f3 x xa | Resumption_corec x \<Rightarrow> f4 x) = f2 x21 x22"
mcont2mcont_llength: fixes lub :: "'a set \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a \<Rightarrow> 'b llist" assumes "mcont lub ord lSup lprefix t" shows "mcont lub ord Sup (\<le>) (\<lambda>x. llength (t x))"
fset_eq_empty_iff: fixes X :: "'a fset" shows "(fset X = {}) = (X = {||})"
cong: fixes \<Sigma> :: "'b list" shows "formula.any \<Sigma> \<equiv> formula.any \<Sigma>"
aj: shows "A $h i $h a \<noteq> (0::'a)"
wf_induct_rule: fixes r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "wf r"   and "\<And>x. (\<And>y. (y, x) \<in> r \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
set_1: fixes x1 :: "('a, 'b, 'c) val" shows "RID\<^sub>E (VE x1) = RID\<^sub>V x1"
Sup_fin_def: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.semilattice_sup sup less_eq less" shows "semilattice_sup.Sup_fin sup \<equiv> semilattice_set.F sup"
store_typing_imp_inst_typing: fixes s :: "s"   and \<S> :: "s_context"   and i :: "nat" assumes "store_typing s \<S>"   and "i < length (inst s)" shows "inst_typing \<S> (inst s ! i) (s_inst \<S> ! i)"
wtc: fixes c :: "'a list"   and pc :: "nat"   and s :: "'a" shows "wtc c pc s = (if c ! pc = \<bottom> then wti c pc s else if s \<sqsubseteq>\<^bsub>r\<^esub> c ! pc then wti c pc (c ! pc) else \<top>)"
insert_Diff1: fixes x :: "'g"   and B :: "'g set"   and A :: "'g set" assumes "x \<in> B" shows "insert x A - B = A - B"
finlsts_rec_LCons: fixes r :: "'b llist"   and A :: "'b set"   and c :: "'c"   and d :: "'b \<Rightarrow> 'b llist \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "finlsts_rec c d (a ## r) = d a r (finlsts_rec c d r)"
monotone_outputs: shows "monotone resumption_ord (\<subseteq>) outputs"
in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright> \<equiv> a \<in> S.domains f \<and> b \<in> S.codomains f"
simps_9: fixes f :: "'g \<Rightarrow> 'f"   and x2 :: "'g" shows "map_option f \<lfloor>x2\<rfloor> = \<lfloor>f x2\<rfloor>"
array_get'_def: fixes a :: "'a array" shows "array_get' a \<equiv> array_get a \<circ> nat_of_integer"
induced_automorphism_eq_foldings_on_chambers_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "fun_eq_on (OpposedThinChamberComplexFoldings.induced_automorphism X f g) g C"
le_max_iff_disj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "less_eq z (ord.max less_eq x y) = (less_eq z x \<or> less_eq z y)"
analz_LtKeys: fixes H :: "msg set" assumes "H \<subseteq> payload \<union> asym.valid \<union> range LtK \<union> Tags" shows "analz H \<inter> range LtK \<subseteq> H"
vars_eval\<^sub>B: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and e :: "'c"   and mem :: "'b \<Rightarrow> 'd"   and v :: "'d" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "x \<notin> bexp_vars e" shows "ev\<^sub>B mem e = ev\<^sub>B (mem(x := v)) e"
ab: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
lbound_def: fixes x :: "real" shows "lbound x = min 0 x"
rep_upd_unit: fixes x :: "nat list"   and n :: "nat"   and i :: "nat"   and a :: "nat" assumes "x = (zeroes n)[i := a]" shows "\<forall>j<length x. (j \<noteq> i \<longrightarrow> x ! j = 0) \<and> (j = i \<longrightarrow> x ! j = a)"
simps_1: shows "Partial_Recursive.list_encode [] = 0"
tab_simps_5: shows "cod ((\<phi> \<star> f) \<cdot> \<rho>) = r' \<star> f"
accmethd_def: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname" shows "accmethd G S C = filter_tab (\<lambda>sig m. G \<turnstile>Methd sig m of C accessible_from S) (methd G C)"
induct: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f"   and "\<And>x. \<lbrakk>x \<in> ccpo.iterates Sup less_eq f; P x\<rbrakk> \<Longrightarrow> P (f x)"   and "\<And>M. \<lbrakk>Complete_Partial_Order.chain less_eq M; \<forall>x\<in>M. x \<in> ccpo.iterates Sup less_eq f \<and> P x\<rbrakk> \<Longrightarrow> P (Sup M)" shows "P x"
distr\<^sub>0_def: fixes ko :: "'c nat_term_order"   and xs :: "('c \<times> 'd) list" shows "distr\<^sub>0 ko xs = MP_oalist (oalist_of_list_ntm (xs, ko))"