hcju/isabelleps huggingface dataset:
value_mI_genTr1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)"   and "j \<le> n" shows "mprod_exp K (K_gamma j) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m_zmax_pdsI K n P I\<^esup> \<in> carrier K"
below_monic_ValKN: shows "below_monic_cfun ValKN"
nfa_reach_is_node: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and p :: "bool list"   and w :: "bool list list"   and q :: "bool list" assumes "wf_nfa A n"   and "nfa_reach A p w q"   and "nfa_is_node A p"   and "list_all (is_alph n) w" shows "nfa_is_node A q"
assms: shows "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>"
h_minus_aux_finite: fixes j :: "nat"   and e :: "'v \<times> 'v" shows "h_minus_aux j e \<noteq> \<top>"
verit_sum_simplify: fixes a :: "'k" shows "a + (0::'k) = a"
dictates2_def: fixes i :: "indi"   and a :: "alt"   and b :: "alt" shows "i dictates a,b \<equiv> i dictates a < b \<and> i dictates b < a"
smult_add_distrib_vec: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec"   and a :: "'a" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "a \<cdot>\<^sub>v (v + w) = a \<cdot>\<^sub>v v + a \<cdot>\<^sub>v w"
dependent_zero: fixes A :: "'c set" assumes "(0::'c) \<in> A" shows "cdependent A"
convert_extTA_simps_3: fixes f :: "'i \<Rightarrow> 'd"   and ta :: "('f, 'c, 'i, 'e, 'g, 'h) thread_action" shows "\<lbrace>convert_extTA f ta\<rbrace>\<^bsub>t\<^esub> = map (convert_new_thread_action f) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"
Lambert_W'_le: fixes x :: "real" shows "Lambert_W' x \<le> - 1"
det_wf_nfa: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat" assumes "subset_DFS A n" shows "wf_dfa (det_nfa A) n"
\<Gamma>_props_6: shows "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
ax_cases: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and C :: "state \<Rightarrow> bool"   and t :: "Term.term"   and Q :: "'b assn" assumes "G,A\<turnstile>{P \<and>. C} t\<succ> {Q}"   and "G,A\<turnstile>{P \<and>. Not \<circ> C} t\<succ> {Q}" shows "G,A\<turnstile>{P} t\<succ> {Q}"
bind_runit: fixes x :: "'a option" shows "Option.bind x Some = x"
pref_prod_pref_short: fixes u :: "'b list"   and z :: "'b list"   and w :: "'b list"   and v :: "'b list" assumes "u \<le>p z \<cdot> w"   and "v \<le>p w"   and "\<^bold>|u\<^bold>| \<le> \<^bold>|z \<cdot> v\<^bold>|" shows "u \<le>p z \<cdot> v"
normalizePrQ'_def: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "normalizePrQ' p = (removeAllDuplicates \<circ> insertDenies \<circ> separate \<circ> qsort' (Nets_List p) \<circ> removeShadowRules2 \<circ> remdups \<circ> rm_MT_rules Cp \<circ> insertDeny \<circ> removeShadowRules1 \<circ> policy2list) p"
assms_3: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M < \<infinity>"
liftAll_def: fixes phi :: "'c \<Rightarrow> bool"   and inp :: "'d \<Rightarrow> 'c option" shows "liftAll phi inp \<equiv> \<forall>i v. inp i = Some v \<longrightarrow> phi v"
if_weak_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'d"   and y :: "'d" assumes "b = c" shows "(if b then x else y) = (if c then x else y)"
to_nat_less_ncols: fixes a :: "'b"   and A :: "(('c, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "mod_type_class.to_nat a < ncols A"
map_snd_list_insert_distrib: fixes x :: "'a \<times> 'b"   and xs :: "('a \<times> 'b) list" assumes "\<forall>(i, p)\<in>insert x (set xs). \<forall>(i', p')\<in>insert x (set xs). p = p' \<longrightarrow> i = i'" shows "map snd (List.insert x xs) = List.insert (snd x) (map snd xs)"
exists_def: shows "exists \<equiv> \<lambda>\<phi>. \<^bold>\<not>(\<^bold>\<forall>x. \<^bold>\<not>\<phi> x)"
abs_eq: fixes x :: "'b \<Rightarrow> 'c option" shows "keys (Mapping x) = dom x"
foo25: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "wellformed_policy3Pr (p @ [x])" shows "wellformed_policy3Pr p"
one_poly_eq_simps_1: shows "(1 = [:1::'c:]) = True"
simps_8: fixes t :: "Syntax.term"   and f :: "(Syntax.var, Syntax.term) fmap" shows "psubst_term (Roll t) f = Roll (psubst_term t f)"
conjI_realizer: fixes P :: "'d \<Rightarrow> bool"   and p :: "'d"   and Q :: "'e \<Rightarrow> bool"   and q :: "'e" assumes "P p"   and "Q q" shows "P (fst (p, q)) \<and> Q (snd (p, q))"
butlast_tl: fixes xs :: "'a list" shows "butlast (tl xs) = tl (butlast xs)"
TESL_interp_stepwise_kills_coind_unfold: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and n :: "nat" shows "\<lbrakk> K\<^sub>1 kills K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = (\<lbrakk> K\<^sub>1 \<not>\<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<union> \<lbrakk> K\<^sub>1 \<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> K\<^sub>2 \<not>\<Up> \<ge> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m) \<inter> \<lbrakk> K\<^sub>1 kills K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
complement_shunting: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and z :: "'c"   and w :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.bounded_distrib_lattice inf less_eq less sup bot top"   and "sup z w = top \<and> inf z w = bot" shows "less_eq (inf z x) y = less_eq x (sup w y)"
fst_Gauss_Jordan_column_k_PA: fixes P :: "(('a, 'c) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and k :: "nat" assumes "P ** B = A" shows "fst (Gauss_Jordan_column_k_PA (P, i, A) k) ** B = snd (snd (Gauss_Jordan_column_k_PA (P, i, A) k))"
degree_power_eq: fixes p :: "'a poly"   and n :: "nat" assumes "p \<noteq> 0" shows "degree (p ^ n) = n * degree p"
ref_reach_unalive: fixes x :: "Value"   and s :: "Store"   and l :: "Location" assumes "\<not> alive x s"   and "s\<turnstile> l reachable_from x" shows "x = ref l"
Up_To_Technique: fixes d :: "'d"   and R :: "(('exp, 'id) MWLfCom list \<times> ('exp, 'id) MWLfCom list) set" assumes "d_Bisimulation_Up_To_USdB d R" shows "R \<subseteq> \<approx>\<^bsub>d\<^esub>"
wf_eq_minimal: fixes r :: "('d \<times> 'd) set" shows "wf r = (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q))"
transfer: shows "rel_fun pcr_word (rel_fun pcr_word pcr_word) (*) (*)"
four_block_mat_dim0: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n 0"   and "C \<in> carrier_mat 0 n"   and "D \<in> carrier_mat 0 0" shows "four_block_mat A B C D = A"
T1: shows "Col A B C \<and> Col A B P \<longrightarrow> (\<exists>Q. A B Perp Q C \<and> Coplanar A B P Q)"
x_def: fixes i :: "nat" shows "local.x i = xs ! i"
next_imp_propagate_next: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" assumes "inv_init_imp_prop_safe c0"   and "InvGlobPointstampsEq c0"   and "cri.InvMsgInGlob (exchange_config c0)"   and "Next' c0 c1" shows "local.next'\<^sup>+\<^sup>+ (prop_config c0 p) (prop_config c1 p)"
rewriteR_comp_comp: fixes g :: "'d \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'd"   and r :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'e" assumes "g \<circ> h = r" shows "f \<circ> g \<circ> h = f \<circ> r"
continuous_on_vector_derivative: fixes S :: "real set"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)" shows "continuous_on S f"
eltp_elt_eq: fixes CC :: "'d set set"   and F :: "'d program" shows "eltp CC F = (\<lambda>x xa. (x, xa) \<in> elt CC F)"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>inpt = inpt r, outpt = outpt r, disk = disk r, dblock = dblock r, phase = phase r, disksWritten = disksWritten r, blocksRead = blocksRead r, allInput = allInput r, chosen = chosen r, \<dots> = state.more r\<rparr>"
the_index_bounded: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "the_index xs x < length xs"
unique_row_col_2: fixes nr1 :: "nat"   and nc1 :: "nat"   and M :: "'b list list"   and nr2 :: "nat"   and nc2 :: "nat" assumes "mat nr1 nc1 M"   and "mat nr2 nc2 M"   and "M \<noteq> []" shows "nc1 = nc2"
cspan_singleton_scaleC: fixes a :: "complex"   and \<psi> :: "'b" assumes "a \<noteq> 0" shows "cspan {a *\<^sub>C \<psi>} = cspan {\<psi>}"
stc_def: fixes x :: "complex star" shows "stc x = (SOME r. x \<in> HFinite \<and> r \<in> SComplex \<and> r \<approx> x)"
simps_1: fixes uu :: "'c"   and uv :: "'c"   and F :: "'a" shows "A2_B.Map (A2_B.MkArr uu uv F) = F"
locally_lipschitz_imp_lipschitz: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and M :: "real" assumes "continuous_on {a..b} f"   and "\<And>x y. \<lbrakk>x \<in> {a..<b}; x < y\<rbrakk> \<Longrightarrow> \<exists>z\<in>{x<..y}. dist (f z) (f x) \<le> M * (z - x)"   and "0 \<le> M" shows "M-lipschitz_on {a..b} f"
is_red_monomial_iff: fixes F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and v :: "'t"   and c :: "'e" shows "is_red F (monomial c v) = (c \<noteq> (0::'e) \<and> (\<exists>f\<in>F. f \<noteq> 0 \<and> lt f adds\<^sub>t v))"
atMostOneClm: fixes ts :: "traffic"   and c :: "cars" shows "|clm ts c| \<le> 1"
assms_1: fixes v :: "'v"   and u :: "'v" assumes "v \<in> V"   and "u \<in> V"   and "v \<noteq> u" shows "\<exists>!n. adjacent v n \<and> adjacent u n"
conf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> subtype P T' T"
convex_scaleR_aux: fixes u :: "real"   and v :: "real"   and x :: "'b" assumes "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R x = x"
simps: fixes e :: "'a"   and a :: "'b"   and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "BsSkewBinomialHeapStruc.elem_invar (BsSkewBinomialHeapStruc.Element e a q) = ((\<forall>x. x \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)) \<longrightarrow> a \<le> BsSkewBinomialHeapStruc.eprio x \<and> BsSkewBinomialHeapStruc.elem_invar x) \<and> SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q))"
\<eta>o_in_hom: fixes y :: "'d" assumes "D.ide y" shows "\<guillemotleft>\<eta>o y : y \<rightarrow>\<^sub>D G (F y)\<guillemotright>"
permutation_bijective: fixes p :: "'c \<Rightarrow> 'c" assumes "permutation p" shows "bij p"
SUP_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b \<times> 'c"   and D :: "'a \<Rightarrow> 'b \<times> 'c" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup_prod (C ` A) = Sup_prod (D ` B)"
sign_simps_12: fixes ac :: "'b"   and bc :: "'b"   and cc :: "'b" shows "ac - bc + cc = ac + cc - bc"
term_f: shows "terminate f (map (\<lambda>g. rec_exec g xs) gs)"
lcos_self: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and H :: "'d set" assumes "Group.group G"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> x <#\<^bsub>G\<^esub> H"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "a \<cdot> b \<in> S"
is_weak_ranking_append: fixes xs :: "'b set list"   and ys :: "'b set list" shows "is_weak_ranking (xs @ ys) = (is_weak_ranking xs \<and> is_weak_ranking ys \<and> set xs \<inter> set ys = {} \<and> \<Union> (set xs) \<inter> \<Union> (set ys) = {})"
funpow_T: fixes i :: "nat" shows "T ^^ i = (\<lambda>f. f \<circ> (\<lambda>n. n + i))"
bind_get: fixes f :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'm"   and g :: "'a \<Rightarrow> 'm" shows "bind (get f) g = get (\<lambda>s. bind (f s) g)"
verts3_slim_in_verts3: fixes v :: "'a" assumes "v \<in> verts3 (with_proj S.slim)" shows "v \<in> verts3 (with_proj slim)"
init_wf_state: fixes r :: "'b rexp" assumes "local.wf n r" shows "wf_state (init r)"
image_chamber_system: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexEndomorphism X f" shows "ChamberComplex.chamber_system (f \<turnstile> X) = f \<turnstile> ChamberComplex.chamber_system X"
ennreal_minus_le_iff: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a - b \<le> c) = (a \<le> b + c \<and> (a = top \<and> b = top \<longrightarrow> c = top))"
equivpartI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" shows "equivpartp r x x"
fps_unit_factor_decompose': fixes f :: "'a fps" shows "f = fps_X ^ subdegree f * unit_factor f"
assms_2: shows "subst_domain \<delta> \<union> range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S"
distinct_4: fixes x31 :: "'c"   and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" shows "Assign x31 x32 \<noteq> Skip"
basis_exists: fixes V :: "(('c \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'c" assumes "V \<subseteq> manifold_eucl.dest.tangent_space p" obtains B :: "(('c \<Rightarrow> real) \<Rightarrow> real) set" where "B \<subseteq> manifold_eucl.dest.tangent_space p" and "B \<subseteq> V" and "independent B" and "V \<subseteq> span B" and "card B = manifold_eucl.dest.tangent_space.dim p V"
bla: fixes i :: "nat" assumes "i \<in> {Suc r..<m}"   and "i \<in> Is p" shows "i \<in> Is q"
invertible_fst_Gauss_Jordan_PA: fixes A :: "(('a, 'c) vec, 'b) vec" shows "invertible (fst (Gauss_Jordan_PA A))"
Iff_Con: fixes p :: "'a pfm"   and p' :: "'a pfm"   and q :: "'a pfm"   and q' :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! p')"   and "\<turnstile>\<^sub>! (q \<^bold>\<longleftrightarrow>\<^sub>! q')" shows "\<turnstile>\<^sub>! (p \<^bold>\<and>\<^sub>! q \<^bold>\<longleftrightarrow>\<^sub>! p' \<^bold>\<and>\<^sub>! q')"
successively_conv_sorted_wrt: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "transp P" shows "successively P xs = sorted_wrt P xs"
prems_5: shows "\<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)"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Decision X"
simps_2: fixes x :: "'b"   and hs :: "'b heap list" shows "del_min (Hp x hs) = pass\<^sub>2 (pass\<^sub>1 hs)"
Vfrom_Ord_bundle_1: fixes A :: "V"   and i :: "V" assumes "A = A"   and "i = i" shows "Vfrom A i = Vfrom A (rank i)"
top_MonoTran_def: shows "\<top> \<equiv> Abs_MonoTran \<top>"
subst_compose: fixes \<sigma> :: "'f \<Rightarrow> ('d, 'g) Term.term"   and \<tau> :: "'g \<Rightarrow> ('d, 'e) Term.term"   and x :: "'f" shows "(\<sigma> \<circ>\<^sub>s \<tau>) x = \<sigma> x \<cdot> \<tau>"
UN_fA: shows "(\<Union>n. fst (A n)) \<in> fmeasurable M"
prf_domain_nonempty: fixes prf_key_gen :: "'a spmf"   and prf_fun :: "'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_domain :: "bool list set"   and prf_dlen :: "nat"   and prf_clen :: "nat"   and upf_key_gen :: "'b spmf" assumes "simple_cipher prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen" shows "prf_domain \<noteq> {}"
cTau: shows "\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'"
is_prefix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "is_prefix xs ys \<equiv> CATCH (ebind (CHECK (length xs \<le> length ys) False) (\<lambda>_. ebind (EWHILEIT (\<lambda>i. i \<le> length xs \<and> take i xs = take i ys) (\<lambda>i. i < length xs) (\<lambda>i. ebind (EASSERT (i < length xs \<and> i < length ys)) (\<lambda>_. ebind (CHECK (xs ! i = ys ! i) False) (\<lambda>_. ERETURN (i + 1)))) 0) (\<lambda>_. THROW True))) ERETURN"
vec_first_def: fixes v :: "'a vec"   and n :: "nat" shows "vec_first v n \<equiv> vec n (($) v)"
insert_update_list: fixes xs :: "(nat \<times> 'd) list"   and a :: "nat"   and b :: "'d"   and x :: "'d" assumes "distinct (map fst xs)"   and "rsorted (map fst xs)"   and "(a, b) \<in> set (update_list a x xs)" shows "b = x"
len: shows "length\<cdot>xs = MkI\<cdot>(n - 1)"
cases: fixes x :: "'a option \<times> 'a list \<times> 'a list" obtains ao :: "'a option" and ys :: "'a list" where "x = (ao__, [], ys__)"   | ao :: "'a option" and xs :: "'a list" where "x = (aoa__, xs__, [])"   | ao :: "'a option" and xa :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = (ao, xa # xs, y # ys)"
nt_sum0: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "a \<in> carrier A" shows "(n + m)\<triangleright>a\<^bsub>A\<^esub> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> m\<triangleright>a\<^bsub>A\<^esub>"
locks_activate_cond_actions2: fixes final1 :: "'f \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and r2 :: "('a, 'b, 'c, 'd, 'e, 'g) semantics"   and bisim :: "'b \<Rightarrow> 'f \<times> 'h \<Rightarrow> 'c \<times> 'd \<Rightarrow> bool"   and \<tau>move2 :: "('a, 'b, 'c, 'd, 'e, 'g) \<tau>moves"   and s1 :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('f \<times> 'a \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and cts :: "'b conditional_action list" shows "locks (FWdelay_bisimulation_base.activate_cond_actions2 final1 final2 r2 bisim \<tau>move2 s1 s2 cts) = locks s2"
fail_nondet_def: fixes return :: "'i \<Rightarrow> 'h"   and empty :: "'i" shows "nondetM_base.fail_nondet return empty = NondetT (return empty)"
that: fixes x :: "'a" assumes "x \<in> cbox a b"   and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)" shows "thesis"
suminf_zero: shows "(\<Sum>n. (0::'c)) = (0::'c)"
pi_approx2: fixes n :: "nat" assumes "even n"   and "0 < n" shows "\<bar>pi - pi_approx2 n\<bar> \<le> inverse (2 ^ (46 * n - 1))"
terminates_Normal_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>Seq c1 c2 \<down> Normal s" obtains "\<Gamma>\<turnstile>c1 \<down> Normal s" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> s'"
cancels_to_1_unfold: fixes x :: "(bool \<times> 'd) list"   and y :: "(bool \<times> 'd) list" assumes "cancels_to_1 x y" obtains xs1 :: "(bool \<times> 'd) list" and x1 :: "bool \<times> 'd" and x2 :: "bool \<times> 'd" and xs2 :: "(bool \<times> 'd) list" where "x = xs1 @ x1 # x2 # xs2" and "y = xs1 @ xs2" and "canceling x1 x2"
c_in_def: shows "c_in = (\<lambda>x u. u div 2 ^ x mod 2)"
inj_of_nat: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_char_0 one times plus zero" shows "inj (semiring_1.of_nat one plus zero)"
Gi: fixes i :: "nat" assumes "i < length F" shows "F ! i \<in> carrier_vec n"
asym: fixes a :: "'f"   and b :: "'f" assumes "a < b"   and "b < a" shows "False"
wf_tuple_def: fixes n :: "nat"   and V :: "nat set"   and x :: "'c option list" shows "wf_tuple n V x = (length x = n \<and> (\<forall>i<n. (x ! i = None) = (i \<notin> V)))"
S_empty: fixes C :: "'a literal multiset" shows "S C = {#}"
set_elem_nth: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "\<exists>m<length xs. xs ! m = x"
lambda_predicates_3_2: fixes F :: "\<Pi>\<^sub>2" shows "[[\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= F]]"
add: fixes b1 :: "'b"   and b2 :: "'b" assumes "b1 \<in> U\<^sub>M\<^sub>_\<^sub>1"   and "b2 \<in> U\<^sub>M\<^sub>_\<^sub>1" shows "f (b1 + b2) = f b1 + f b2"
dickson_grading_varnum: fixes X :: "'b set" assumes "finite X" shows "dickson_grading (varnum X)"
mset_le_trans: fixes K :: "'a multiset"   and M :: "'a multiset"   and N :: "'a multiset" assumes "K < M"   and "M < N" shows "K < N"
r_nf_u_prim: shows "prim_recfn 1 r_nf_u"
cbelow_on_1_def: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "cbelow_on_1 A ord z rel \<equiv> (relbelow_on A ord z rel)\<^sup>+\<^sup>+"
Rdual: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<forall>S\<subseteq>A. \<exists>L. isLub S \<lparr>pset = A, order = r\<rparr> L" shows "\<forall>S\<subseteq>A. \<exists>G. isGlb S \<lparr>pset = A, order = r\<rparr> G"
coll_minus_eq: fixes x :: "real \<times> real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "coll (x - a) (x - b) (x - c) = coll a b c"
WbisT_MtransC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>wT d"   and "s \<approx> t"   and "(c, s) \<rightarrow>*c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t')" and "s' \<approx> t'" and "c' \<approx>wT d'"
image_insert: fixes f :: "'g \<Rightarrow> 'f"   and a :: "'g"   and B :: "'g set" shows "f ` insert a B = insert (f a) (f ` B)"
HoVH_def: fixes \<mu> :: "'b \<times> 'b \<times> 'b" shows "HoVH \<mu> \<equiv> if VVV.arr \<mu> then fst \<mu> \<star> fst (snd \<mu>) \<star> snd (snd \<mu>) else null"
self_redundant_clause: fixes C :: "'a eclause"   and S :: "'a eclause set"   and C' :: "'a literal set"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "C \<in> S"   and "C' = cl_ecl C"   and "ground_clause (subst_cl (cl_ecl C) \<sigma>)" shows "redundant_clause (subst_ecl C \<sigma>) S \<sigma> C'"
VSGeneral: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSGeneral \<phi>) xs = eval \<phi> xs"
ladder_every_stage_4: fixes index :: "nat" assumes "index < length L" shows "ladder_\<alpha> (\<alpha> @ \<delta>) D L index = ladder_\<alpha> \<alpha> E L' index @ \<delta>"
topspace_map_topology: fixes f :: "'d \<Rightarrow> 'c"   and T :: "'d topology" shows "topspace (map_topology f T) = f ` topspace T"
is_lasso_pre_def: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and L :: "('Q \<times> nat) lasso" shows "graph.is_lasso_pre (degeneralize_ext m) L \<equiv> lasso_v0 L \<in> g_V0 (degeneralize_ext m) \<and> path (g_E (degeneralize_ext m)) (lasso_v0 L) (lasso_reach L) (lasso_va L) \<and> path (g_E (degeneralize_ext m)) (lasso_va L) (lasso_cycle L) (lasso_va L)"
shift_snth: fixes xs :: "'b list"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "(xs @- \<sigma>) i = (if i < length xs then xs ! i else \<sigma> (i - length xs))"