hcju/isabelleps huggingface dataset:
Zconseq': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>Z {P} c {Q}"   and "\<And>s. P s \<le> P' s"   and "\<And>s. Q' s \<le> Q s" shows "\<turnstile>\<^sub>Z {P'} c {Q'}"
iso_digraph_axioms: shows "\<AA> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
mojmir_accept_token_set_def2: fixes i :: "nat" assumes "finite fail"   and "finite (local.merge i)"   and "infinite (succeed i)" shows "accept\<^sub>M"
simps_205: fixes x15 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "throw x15 \<noteq> Call x101 x102 x103 x104"
reachable_n: fixes A :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and s :: "'e"   and a :: "'d"   and t :: "'e" assumes "reachable A s"   and "(s, a, t) \<in> trans_of A" shows "reachable A t"
maximal: fixes \<aa> :: "('a \<times> 'a) set set" assumes "lideal \<aa> carrier_local_ring_at add_local_ring_at mult_local_ring_at zero_local_ring_at one_local_ring_at"   and "\<aa> \<noteq> carrier_local_ring_at" shows "\<aa> \<subseteq> \<ww>"
cet_int: fixes t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL" shows "\<Gamma> \<turnstile>\<^sub>c \<integral>\<^sub>c e \<partial>t : REAL"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "A.map f = C.null"
fresh_star_set: fixes xs :: "'c set"   and c :: "'d" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>xs. x \<sharp> c"
span_span: fixes W :: "'a vec set"   and n :: "nat" assumes "W \<subseteq> carrier_vec n" shows "LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (LinearCombinations.module.span class_ring (module_vec TYPE('a) n) W) = LinearCombinations.module.span class_ring (module_vec TYPE('a) n) W"
unique_factorization_def: fixes Factors :: "'a poly set"   and f :: "'a poly"   and cfs :: "'a \<times> 'a poly multiset" shows "unique_factorization Factors f cfs = (Collect (factorization Factors f) = {cfs})"
det_four_block_mat_upper_right_zero: fixes A1 :: "'b mat"   and n :: "nat"   and A2 :: "'b mat"   and m :: "nat"   and A3 :: "'b mat"   and A4 :: "'b mat" assumes "A1 \<in> carrier_mat n n"   and "A2 = 0\<^sub>m n m"   and "A3 \<in> carrier_mat m n"   and "A4 \<in> carrier_mat m m" shows "det (four_block_mat A1 A2 A3 A4) = det A1 * det A4"
unit_def: fixes a :: "'a arrow_of_spans_data" shows "\<i>[a] \<equiv> \<lparr>Chn = \<p>\<^sub>0[Chn a, Chn a], Dom = Dom (a \<star> a), Cod = Cod a\<rparr>"
set_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "lset (LCons x21 x22) = insert x21 (lset x22)"
prems: shows "\<phi> \<sim>\<^sub>C \<psi>1 or\<^sub>n \<psi>2"
prod_dfs_inj: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list"   and n :: "nat"   and x :: "nat \<times> nat"   and i1 :: "nat"   and i2 :: "nat"   and j1 :: "nat"   and j2 :: "nat"   and k :: "nat" assumes "prod_DFS A B n"   and "prod_is_node A B x"   and "dfa_is_node A i1"   and "dfa_is_node B i2"   and "dfa_is_node A j1"   and "dfa_is_node B j2"   and "fst (prod_dfs A B x) ! i1 ! i2 = Some k"   and "fst (prod_dfs A B x) ! j1 ! j2 = Some k" shows "(i1, i2) = (j1, j2)"
primes_omega_prime: fixes p :: "nat" assumes "prime p" shows "primes_omega p = 1"
Domain_def: shows "Domain \<equiv> \<lambda>r. {x. Domainp (\<lambda>x xa. s2p r (x, xa)) x}"
lossless_spmfD_set_spmf_nonempty: fixes p :: "'a spmf" assumes "lossless_spmf p" shows "set_spmf p \<noteq> {}"
bij_betw_arr_sets: shows "bij_betw \<psi> (Collect arr) (Collect C'.arr)"
c_assoc_have_key_df: fixes y :: "nat"   and x :: "nat" shows "c_assoc_have_key y x = (if y = 0 then 1 else if c_fst (c_hd y) = x then 0 else c_assoc_have_key (c_tl y) x)"
ZFTriple_def: fixes a :: "ZF"   and b :: "ZF"   and c :: "ZF" shows "ZFTriple a b c = Opair (Opair a b) c"
numeral_code_3: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = (let m = numeral.numeral one plus n in plus (plus m m) one)"
ide_in_hom: fixes a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "A_B.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] a\<guillemotright>"
take_non_empty: fixes xs :: "int list" assumes "xs \<noteq> []" shows "Challenge1A.take xs \<noteq> []"
fully_unsorted_poly_rel_vars_subset_vars_llist: fixes A :: "(char list list \<times> int) list"   and B :: "int mpoly" assumes "(A, B) \<in> fully_unsorted_poly_rel O mset_poly_rel" shows "vars B \<subseteq> \<phi> ` vars_llist A"
list_of_oalist_tc_inject: fixes x :: "('a, 'b) oalist_tc"   and y :: "('a, 'b) oalist_tc" shows "(list_of_oalist_tc x = list_of_oalist_tc y) = (x = y)"
emeasure_bounded_finite: fixes A :: "'b set" assumes "bounded A" shows "emeasure lborel A < \<infinity>"
dual_meet: fixes x :: "'c"   and y :: "'c" shows "dual (x \<sqinter> y) = dual x \<squnion> dual y"
h2: shows "insertion (nth_default 0 (L[var := x])) p = insertion (nth_default 0 (L[var := x])) (\<Sum>i\<le>d. isolate_variable_sparse p var i * Var var ^ i)"
perm_restrict_simps_2: fixes x :: "'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "x \<notin> S" shows "perm_restrict f S x = x"
simps_3: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "'a"   and va :: "'a list" shows "PAC_Checker_Init.merge f [] (v # va) = v # va"
frag_cmul_minus_one: fixes x :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul (- 1) x = - x"
llength_ldropn: fixes n :: "nat"   and xs :: "'a llist" shows "llength (ldropn n xs) = llength xs - enat n"
disjoint_insert_2: fixes A :: "'e set"   and b :: "'e"   and B :: "'e set" shows "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
ln_lower_15_def: shows "ln_lower_15 \<equiv> \<lambda>x. - ln_upper_15 (inverse x)"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc (C\<^sub>1;; C\<^sub>2) Q = (vc C\<^sub>1 (pre C\<^sub>2 Q) \<and> vc C\<^sub>2 Q)"
listEqualsButlastAppendLast: fixes list :: "'a list" assumes "list \<noteq> []" shows "list = butlast list @ [last list]"
add_eq_0_iff: fixes x :: "V"   and y :: "V" shows "(x + y = 0) = (x = 0 \<and> y = 0)"
Lit: fixes G :: "prog"   and s :: "st"   and v :: "val"   and n :: "nat" shows "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<midarrow>n\<rightarrow> Norm s"
hom_zero: fixes d :: "nat" shows "monom (0::'a) d = 0"
simps_2: fixes u :: "'b trm"   and c :: "'b" shows "u \<prec> Const c = False"
longest_Lyndon_suf_max: fixes v :: "'a list"   and w :: "'a list" assumes "v \<le>s w"   and "Lyndon v" shows "v \<le>s LynSuf w"
oLimit_eqI: fixes f :: "nat \<Rightarrow> ordinal"   and g :: "nat \<Rightarrow> ordinal" assumes "\<And>n. \<exists>m. f n \<le> g m"   and "\<And>n. \<exists>m. g n \<le> f m" shows "oLimit f = oLimit g"
mono: fixes joint_oracle :: "'f \<Rightarrow> 'g \<Rightarrow> 'd \<Rightarrow> (('e option \<times> 'f) option \<times> ('e option \<times> 'g) option) pmf"   and callee1 :: "'f \<Rightarrow> 'd \<Rightarrow> ('e option \<times> 'f) spmf"   and x :: "((('b option, 'd, 'e) gpv \<times> 'f) \<times> 'g) \<times> bool" shows "spmf.mono_body (\<lambda>exec_until_stop. case x of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> (case x of (gpv, s1) \<Rightarrow> \<lambda>s2 b. if b then the_gpv gpv \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (x, s1, s2) | IO out rpv \<Rightarrow> joint_oracle s1 s2 out \<bind> (\<lambda>(a, b). case a of None \<Rightarrow> return_pmf None | Some (r1, s1') \<Rightarrow> case b of Some (r2, s2') \<Rightarrow> case (r1, r2) of (None, None) \<Rightarrow> curry (curry (curry exec_until_stop)) (Generative_Probabilistic_Value.Done None) s1' s2' True | (None, _) \<Rightarrow> curry (curry (curry exec_until_stop)) (Generative_Probabilistic_Value.Done None) s1' s2' True | (Some r1', None) \<Rightarrow> curry (curry (curry exec_until_stop)) (rpv r1') s1' s2' False | (Some r1', Some r2') \<Rightarrow> curry (curry (curry exec_until_stop)) (rpv r1') s1' s2' True)) else the_gpv gpv \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (None, s1, s2) | IO out rpv \<Rightarrow> callee1 s1 out \<bind> (\<lambda>(r1, s1'). case r1 of None \<Rightarrow> curry (curry (curry exec_until_stop)) (Generative_Probabilistic_Value.Done None) s1' s2 False | Some r1' \<Rightarrow> curry (curry (curry exec_until_stop)) (rpv r1') s1' s2 False))) xa) xa)"
Digamma_def: fixes z :: "'a" shows "Digamma z = (\<Sum>k. inverse (of_nat (Suc k)) - inverse (z + of_nat k)) - euler_mascheroni"
assms_2: shows "\<HH> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
that: fixes g :: "complex \<Rightarrow> complex" assumes "g holomorphic_on A"   and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x" shows "thesis"
bst_union: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.bst (Set2_Join.union join t1 t2)"
Mf_def: fixes m :: "int"   and cfs :: "int \<times> int poly multiset" shows "poly_mod.Mf m cfs \<equiv> case cfs of (c, fs) \<Rightarrow> (poly_mod.M m c, image_mset (poly_mod.Mp m) fs)"
a: shows "\<turnstile> (\<Delta> @ TVarB X T # \<Gamma>) ok"
is_prefix_butlast: fixes q :: "'a list"   and p :: "'a list" assumes "is_prefix q (butlast p)" shows "is_prefix q p"
plus_1: fixes x :: "'a" shows "frac (x + (1::'a)) = frac x"
isomorphic_implies_equicomposable_2: fixes f :: "'a"   and g :: "'a"   and \<sigma> :: "'a" assumes "f \<cong> g" shows "(f \<star> \<sigma> \<noteq> null) = (g \<star> \<sigma> \<noteq> null)"
Bernstein_Poly_01_def: fixes j :: "nat"   and p :: "nat" shows "Bernstein_Poly_01 j p = monom (real (p choose j)) j * monom 1 (p - j) \<circ>\<^sub>p [:1, - 1:]"
valid_unMultigraph_def: fixes G :: "('a, 'b) graph" shows "valid_unMultigraph G \<equiv> valid_graph G \<and> valid_unMultigraph_axioms G"
exposed_face_of_Inter: fixes P :: "'b set set"   and S :: "'b set" assumes "P \<noteq> {}"   and "\<And>T. T \<in> P \<Longrightarrow> T exposed_face_of S" shows "\<Inter> P exposed_face_of S"
refines_def: fixes R :: "('a \<times> 'b) set"   and pi :: "'c \<Rightarrow> 'd"   and Sa :: "('a, 'd) spec"   and Sc :: "('b, 'c) spec" shows "refines R pi Sa Sc \<equiv> obs_consistent R pi Sa Sc \<and> PO_refines R Sa Sc"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_plus" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> vint_plus). vint_plus\<lparr>a\<rparr> = b"
vec_to_iarray_least_non_zero_position_of_vector_from_index': fixes i :: "'c"   and j :: "'d"   and A :: "(('b, 'd) vec, 'c) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "least_non_zero_position_of_vector_from_index (vec_to_iarray (column j A)) (mod_type_class.to_nat i) = mod_type_class.to_nat (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n)"
dc_crypto_def: shows "dc_crypto = dining_cryptographers - {None} \<times> UNIV"
prefix_to_carriers: fixes xs :: "'msg event list"   and i :: "nat" assumes "xs prefix of i" shows "set xs \<subseteq> set (history i)"
countable_support: shows "countable {x. prob {x} \<noteq> 0}"
assms_1: shows "isotone (\<lambda>x. \<nu> f x)"
subgroup_generated_is_subset: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and H :: "'c set" assumes "Group_Theory.group G composition unit" shows "group.subgroup_generated G composition unit H \<subseteq> G"
State_Idle_imp_exists_state_change: fixes localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and n :: "nat" assumes "\<not> State_Idle localState output_fun trans_fun (localState c)"   and "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c))" shows "\<exists>i<n. \<not> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>i\<^esup> c)) \<and> (\<forall>j\<le>n. i < j \<longrightarrow> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>j\<^esup> c)))"
itrm_bracket_opaque: fixes i :: "nat"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_bracket i x = Some x'" shows "set (opaque x') = set (opaque x) - {i}"
flow_initial_time: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and t0 :: "real"   and x0 :: "'a" assumes "ll_on_open_it T f X"   and "t0 \<in> T"   and "x0 \<in> X" shows "ll_on_open.flow T f X t0 x0 t0 = x0"
not_Interp_imp_general: fixes D' :: "'a literal multiset"   and D :: "'a literal multiset"   and C :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "D' \<le> D"   and "D' \<in> N \<and> S D' = {#} \<or> Max (atms_of D') < Max (atms_of D)"   and "\<not> Interp D \<Turnstile> D'"   and "D < C"   and "interp C \<subseteq> \<Union> (production ` CC)" shows "\<not> \<Union> (production ` CC) \<Turnstile> D'"
hv_props_7: shows "\<forall>(k, \<gamma>)\<in>\<H>. valid_path \<gamma>"
diamond_edges_are_valid_3: shows "valid_path (snd diamond_bot_left_edge)"
spl: shows "zsplit0 a = (fst (zsplit0 a), snd (zsplit0 a))"
coset_join1: fixes H :: "'a set"   and x :: "'a" assumes "H #>\<^bsub>add_monoid G\<^esub> x = H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "x \<in> H"
upres: shows "mupdate i a t = binsert fst (i, a) t"
authK_authentic: fixes A :: "agent"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key authK, Peer, Ta\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbV" shows "\<exists>AT. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Peer, Ta\<rbrace>, AT\<rbrace> \<in> set evs"
exhaust: fixes y :: "recf" obtains "y = Z"   | "y = S"   | x31 :: "nat" and x32 :: "nat" where "y = recf.Id x31 x32"   | x41 :: "nat" and x42 :: "recf" and x43 :: "recf list" where "y = Cn x41 x42 x43"   | x51 :: "nat" and x52 :: "recf" and x53 :: "recf" where "y = Pr x51 x52 x53"   | x61 :: "nat" and x62 :: "recf" where "y = Mn x61 x62"
cases: fixes x :: "'a ext0" obtains "x = ext0.Bot"   | uu_ :: "'a" where "x = ext0.Val uu_"   | "x = ext0.Top"
unrest_sublens: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "('c, 'b) uexpr"   and y :: "'d \<Longrightarrow> 'b" assumes "x \<sharp> P"   and "y \<subseteq>\<^sub>L x" shows "y \<sharp> P"
happen_implies_can_occur: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration" assumes "c \<turnstile> ev \<mapsto> c'" shows "can_occur ev c"
ctr_transfer_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun R (rel_option R) pure pure"
interleaves_nil: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and ys :: "'b list"   and zs :: "'b list" assumes "[] \<simeq> {ys, zs, P}" shows "ys = [] \<and> zs = []"
alw_destL: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "alw \<phi> \<pi>l" shows "\<phi> \<pi>l"
mset_neg_plus: fixes A :: "'b zmultiset"   and B :: "'b zmultiset" shows "mset_neg (A + B) = mset_neg A - mset_pos B + (mset_neg B - mset_pos A)"
triv_rel: fixes A :: "'a set" shows "(\<lambda>_\<in>A. 0) \<in> relations A"
sub_def: shows "Code_Numeral.sub \<equiv> map_fun id (map_fun id integer_of_int) (\<lambda>m n. numeral m - numeral n)"
part_product_identity: fixes x :: "i"   and y :: "i" assumes "CM"   and "P x y" shows "x \<otimes> y = x"
entails_frame: fixes P :: "'b \<Rightarrow> real"   and R :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and S :: "'b \<Rightarrow> real" assumes "P \<tturnstile> R"   and "Q \<tturnstile> S" shows "P && Q \<tturnstile> R && S"
L_\<beta>: shows "Limit (\<Union>\<^sub>\<circ> (set {\<gamma>. Limit \<gamma> \<and> \<gamma> \<subseteq>\<^sub>\<circ> \<alpha>}))"
drop_map: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "drop n (map f xs) = map f (drop n xs)"
lookup_permutep: fixes f :: "'c \<Rightarrow> 'c"   and m :: "'c \<Rightarrow>\<^sub>0 'b"   and x :: "'c" shows "lookup (permutep f m) x = (if bij f then lookup m (f x) else lookup m x)"
matchs_mono: fixes fps :: "(char list \<times> pat) list"   and fs :: "(char list \<times> trm) list"   and ts :: "trm list"   and l :: "char list"   and t :: "trm" assumes "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts"   and "fps\<langle>l\<rangle>\<^sub>? = \<bottom>" shows "\<turnstile> fps [\<rhd>] (l, t) \<Colon> fs \<Rightarrow> ts"
parallel_converter2_map1_out: fixes f :: "'a \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'c"   and h :: "'k \<Rightarrow> 'e"   and k :: "'g \<Rightarrow> 'l"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and conv2 :: "('b, 'd, 'f, 'h) converter" shows "map_converter f g h k conv1 |\<^sub>= conv2 = map_converter (map_sum f id) (map_sum g id) (map_sum h id) (map_sum k id) (conv1 |\<^sub>= conv2)"
F_in_rell: fixes L1 :: "'n \<Rightarrow> 'x \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'y \<Rightarrow> bool"   and L3 :: "'p \<Rightarrow> 'z \<Rightarrow> bool" shows "rell_F L1 L2 L3 = (\<lambda>x y. \<exists>z. (set1_F z \<subseteq> {(x, y). L1 x y} \<and> set2_F z \<subseteq> {(x, y). L2 x y} \<and> set3_F z \<subseteq> {(x, y). L3 x y}) \<and> mapl_F fst fst fst z = x \<and> mapl_F snd snd snd z = y)"
bern_eq: shows "Bernstein_coeffs p c d [:b, a:] = map (\<lambda>i. a * (real i * d + real (p - i) * c) / real p + b) [0..<p + 1]"
polyhedron_Int_affine_parallel: fixes S :: "'b set" shows "polyhedron S = (\<exists>F. finite F \<and> S = affine hull S \<inter> \<Inter> F \<and> (\<forall>h\<in>F. \<exists>a b. a \<noteq> (0::'b) \<and> h = {x. a \<bullet> x \<le> b} \<and> (\<forall>x\<in>affine hull S. x + a \<in> affine hull S)))"
l_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(x \<otimes> y = x \<otimes> z) = (y = z)"
lea_total: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F \<or> Tarski_neutral_dimensionless.LeA Bet Cong D E F A B C"
WTrt_binop_simps_5: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>GreaterThan\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Boolean)"
node_step_no_change_on_send_or_receive: fixes \<sigma> :: "nat \<Rightarrow> state"   and i :: "nat"   and P :: "(state, msg, pseqp, pseqp label) seqp \<times> msg list \<times> (msg list, msg, unit, unit label) seqp"   and R :: "nat set"   and a :: "msg node_action"   and \<sigma>' :: "nat \<Rightarrow> state"   and i' :: "nat"   and P' :: "(state, msg, pseqp, pseqp label) seqp \<times> msg list \<times> (msg list, msg, unit, unit label) seqp"   and R' :: "nat set" assumes "((\<sigma>, NodeS i P R), a, \<sigma>', NodeS i' P' R') \<in> onode_sos (oparp_sos i (oseqp_sos \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G))"   and "a \<noteq> \<tau>" shows "\<sigma>' i = \<sigma> i"
compl_sup_top: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup (uminus x) x = top"
mkArr_Fun: fixes f :: "'s" assumes "arr f" shows "mkArr (local.set (local.dom f)) (local.set (cod f)) (Fun f) = f"
RS_simps_19: shows "Maps.Dom rs.S\<^sub>1 = src (tab\<^sub>0 s)"
snapshot_state_unchanged_trace: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and p :: "'p"   and u :: "'s" assumes "trace c t c'"   and "ps c p = Some u" shows "ps c' p = Some u"
wrapH_unwrapH_id: shows "wrapH oo unwrapH = ID"
strict_trans1: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "b \<sqsubseteq> a"   and "c \<sqsubset> b" shows "c \<sqsubset> a"
simps_155: fixes x31 :: "('a, 'b, 'c) expr"   and x32 :: "('a, 'b, 'c) expr"   and x33 :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (Ite x31 x32 x33) = RID\<^sub>E x31 \<union> RID\<^sub>E x32 \<union> RID\<^sub>E x33"
hyps: fixes A :: "(('a, 'n) vec, 'm) vec"   and P :: "(('a, 'm) vec, 'm) vec" assumes "n < nrows A"   and "invertible P" shows "invertible (fst (row_add_iterate_PA (P, A) n i j))"
continuous_at_iff_ereal: fixes x0 :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> real" shows "continuous (at x0 within s) f = continuous (at x0 within s) (ereal \<circ> f)"
finite_subcls1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "finite (subcls1 P)"
ccpo2: shows "class.ccpo lubb ordb (mk_less ordb)"
Sup_range_enat: shows "\<Squnion> range enat = \<infinity>"
card_ge_0_finite: fixes A :: "'a set" assumes "0 < card A" shows "finite A"
preserves_hom: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.VV.in_hom f a b" shows "\<guillemotleft>fst f \<star>\<^sub>C snd f : fst a \<star>\<^sub>C snd a \<Rightarrow>\<^sub>C fst b \<star>\<^sub>C snd b\<guillemotright>"
finite_measure_Diff': fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M (A \<inter> B)"
simps_1: fixes P :: "'a" shows "post (SKIP {P}) = P"
scaleR_minus_left: fixes a :: "real"   and x :: "'b" shows "- a *\<^sub>R x = - (a *\<^sub>R x)"
useNodes_of_def: fixes \<alpha>n :: "'c \<Rightarrow> 'b list"   and "uses" :: "'c \<Rightarrow> 'b \<Rightarrow> 'a set"   and g :: "'c" shows "CFG_base.useNodes_of \<alpha>n uses g = fold (CFG_base.addN uses g) (\<alpha>n g) Mapping.empty"
inter_filter: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "finite A" shows "sum g {x \<in> A. P x} = (\<Sum>x\<in>A. if P x then g x else (0::'b))"
r_e2xs_def: shows "r_e2xs \<equiv> Cn 1 r_pdec12 [r_e2frame]"
diff_add_assoc2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b + c - a = b - a + c"
expand_superset: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "{a. g a \<noteq> (0::'d)} \<subseteq> A" shows "Sum_any g = sum g A"