hcju/isabelleps huggingface dataset:
listindices_aux: fixes i :: "nat"   and V :: "'a list" assumes "i < length V" shows "V ! i \<in> set V"
strong_induct: fixes x :: "fm"   and P :: "'a \<Rightarrow> fm \<Rightarrow> bool"   and c :: "'a" assumes "ss_fm x"   and "\<And>i j c. P c (Var i IN Var j)"   and "\<And>A B c. \<lbrakk>ss_fm A; \<And>c. P c A; ss_fm B; \<And>c. P c B\<rbrakk> \<Longrightarrow> P c (A OR B)"   and "\<And>A B c. \<lbrakk>ss_fm A; \<And>c. P c A; ss_fm B; \<And>c. P c B\<rbrakk> \<Longrightarrow> P c (A AND B)"   and "\<And>A i c. \<lbrakk>{atom i} \<sharp>* c; ss_fm A; \<And>c. P c A\<rbrakk> \<Longrightarrow> P c (SyntaxN.Ex i A)"   and "\<And>A j i c. \<lbrakk>{atom i} \<sharp>* c; ss_fm A; \<And>c. P c A; atom j \<sharp> (i, A)\<rbrakk> \<Longrightarrow> P c (All2 i (Var j) A)" shows "P c x"
apply_cm_preserves_cond: fixes \<Delta> :: "('a \<times> 't) zmultiset"   and c0 :: "('a, 't, 'b) Propagate.configuration_scheme"   and loc' :: "'a"   and t'' :: "'t"   and n :: "int" assumes "\<forall>(loc, t)\<in>#\<^sub>z\<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t" shows "\<forall>(loc, t)\<in>#\<^sub>z\<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp (apply_cm c0 loc' t'' n) loc) \<and> t' \<le> t"
eq_numeral_iff_iszero_10: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(zero = numeral.numeral one plus y) = ring_1.iszero zero (numeral.numeral one plus y)"
n_subsets: fixes A :: "'a set"   and k :: "nat" assumes "finite A" shows "card {B. B \<subseteq> A \<and> card B = k} = card A choose k"
lookup_filter: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and t :: "('b, 'c) mapping_rbt"   and k :: "'b" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.lookup (RBT_Mapping2.filter P t) k = (case RBT_Mapping2.lookup t k of None \<Rightarrow> None | Some v \<Rightarrow> if P (k, v) then Some v else None)"
lunit_coherence: fixes f :: "'c" assumes "C.ide f" shows "\<l>\<^sub>D[H f] = H \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi>\<^sub>H (trg\<^sub>C f, f) \<cdot>\<^sub>D (H.unit (trg\<^sub>C f) \<star>\<^sub>D H f)"
adjunction_in_normal_bicategory_axioms: shows "adjunction_in_normal_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<zeta> \<xi>"
rsplit_def: fixes f :: "int \<Rightarrow> MIR.num \<Rightarrow> fm"   and a :: "MIR.num" shows "rsplit f a \<equiv> foldr MIR.disj (map (\<lambda>(\<phi>, n, s). MIR.conj \<phi> (f n s)) (rsplit0 a)) F"
wf_idecl_def: fixes G :: "prog" shows "wf_idecl G = (\<lambda>(I, i). ws_idecl G I (isuperIfs i) \<and> \<not> is_class G I \<and> (\<forall>(sig, mh)\<in>set (imethods i). wf_mhead G (pid I) sig mh \<and> \<not> is_static mh \<and> accmodi mh = Public) \<and> unique (imethods i) \<and> (\<forall>J\<in>set (isuperIfs i). is_acc_iface G (pid I) J) \<and> (table_of (imethods i) hiding methd G Object under \<lambda>new old. accmodi old \<noteq> Private entails \<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old \<and> is_static new = is_static old) \<and> (set_option \<circ> table_of (imethods i) hidings Un_tables (DeclConcepts.imethds G ` set (isuperIfs i)) entails \<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old))"
p_on_line_def: shows "p_on_line \<equiv> TarskiAbsolute.on_line p_between"
simps: fixes a1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and a2 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" shows "a1 \<hookrightarrow>\<^sub>i a2 = (\<exists>\<Gamma> n \<Phi>. a1 = \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<and> a2 = \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [])"
real_sqrt_has_derivative_generic: fixes x :: "real"   and S :: "real set" assumes "x \<noteq> 0" shows "(sqrt has_derivative (*) ((if 0 < x then 1 else - 1) * inverse (sqrt x) / 2)) (at x within S)"
sel_8: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.UBI_upd (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x8"
distinct_61: fixes x4 :: "ltkey"   and x5 :: "ephkey" shows "cLtK x4 \<noteq> cEphK x5"
E_prod_def: shows "E = {x. snd x \<in> E}"
map_image_filter_axioms: shows "map_image_filter \<alpha>1 invar1 \<alpha>2 invar2 map_image_filter"
\<pi>_nth_prime: fixes n :: "nat" shows "\<pi> (real (nth_prime n)) = real n + 1"
iso_kernel_image: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "Group.group H" shows "(f \<in> Group.iso G H) = (f \<in> hom G H \<and> kernel G H f = {\<one>\<^bsub>G\<^esub>} \<and> f ` carrier G = carrier H)"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "('test, 'atom) com \<times> 'state" assumes "a1 \<rightarrow>*c a2" obtains c :: "('test, 'atom) com" and s :: "'state" where "a1 = (c__, s__)" and "a2 = (c__, s__)"   | c :: "('test, 'atom) com" and s :: "'state" and c' :: "('test, 'atom) com" and s' :: "'state" and c'' :: "('test, 'atom) com" and s'' :: "'state" where "a1 = (c, s)" and "a2 = (c'', s'')" and "(c, s) \<rightarrow>*c (c', s')" and "(c', s') \<rightarrow>c (c'', s'')"
convex_hull_insert: fixes S :: "'c set"   and a :: "'c" assumes "S \<noteq> {}" shows "convex hull insert a S = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. u + v = 1 \<and> b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b}"
sub_def: fixes as :: "'a iarray"   and n :: "nat" shows "as !! n = IArray.list_of as ! n"
assms_3: shows "(x2, K2) \<in> \<D>"
measurable_restrict_proj: fixes J :: "'d set"   and II :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" assumes "J \<in> II"   and "finite J" shows "finmap_of J \<in> Pi\<^sub>M J M \<rightarrow>\<^sub>M Pi\<^sub>F II M"
vars_vec_append_subset: fixes n :: "nat"   and v :: "rat vec" shows "vars (vec_to_lpoly (0\<^sub>v n @\<^sub>v v)) \<subseteq> {n..<n + dim_vec v}"
assms: shows "P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'"
AbstrLevels_A2_A22: fixes i :: "AbstrLevelsID" assumes "sA2 \<in> AbstrLevel i" shows "sA22 \<notin> AbstrLevel i"
simps_1: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and signalsInit2 :: "signal \<Rightarrow> nat"   and signalsResp2 :: "signal \<Rightarrow> nat"   and chan :: "chan set"   and bad :: "agent set"   and more :: "'a" shows "chan \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, signalsInit2 = signalsInit2, signalsResp2 = signalsResp2, chan = chan, bad = bad, \<dots> = more\<rparr> = chan"
ordered_nsets_4_eq: fixes A :: "'a set" shows "[A]\<^bsup>4\<^esup> = {{u, x, y, z} |u x y z. u \<in> A \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> u < x \<and> x < y \<and> y < z}"
k_1: shows "2 powr real_of_int k \<le> real_of_int \<bar>ai\<bar>"
the_var_FVar_def: fixes G :: "prog"   and s :: "state"   and accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and a :: "expr"   and fn :: "vname" shows "the_var G s ({accC,statDeclC,stat}a..fn) = fvar statDeclC stat fn (the_val a) s"
even_take_bit_eq: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and n :: "nat"   and a :: "'k" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (take_bit n a) = (n = 0 \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
simps_5: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" shows "usubstappt \<sigma> U (Plus \<theta> \<eta>) = Pluso (usubstappt \<sigma> U \<theta>) (usubstappt \<sigma> U \<eta>)"
f: fixes x :: "'a" assumes "x \<in> set_spmf p" shows "rel_spmf R (f x) q"
add_0_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.monoid_add plus zero" shows "plus zero a = a"
type_wfttype_inhabited: fixes \<tau> :: "('fun, 'atom) Term.term" assumes "\<And>f T. Fun f T \<sqsubseteq> \<tau> \<Longrightarrow> 0 < arity f"   and "wf\<^sub>t\<^sub>r\<^sub>m \<tau>" shows "\<exists>t. \<Gamma> t = \<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m t"
Diagonalize_Tensor: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Arr t"   and "Arr u" shows "\<^bold>\<lfloor>t \<^bold>\<otimes> u\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<otimes> \<^bold>\<lfloor>u\<^bold>\<rfloor>\<^bold>\<rfloor>"
simps_24: fixes x51 :: "bexp"   and x52 :: "com"   and x41 :: "bexp"   and x42 :: "com"   and x43 :: "com" shows "WHILE x51 DO x52 \<noteq> IF x41 THEN x42 ELSE x43"
naturality: fixes f :: "'a" assumes "R.arr f" shows "R.comp (R.map (R.cod f)) (R.map f) = R.comp (R.map f) (R.map (R.dom f))"
sets_eq: shows "sets (dens_ctxt_measure ({}, V', \<Gamma>, \<lambda>_. 1) \<rho> \<bind> f) = sets (f \<rho>)"
insert_in_sets: fixes x :: "'c"   and M :: "'c measure"   and A :: "'c set" assumes "{x} \<in> sets M"   and "A \<in> sets M" shows "insert x A \<in> sets M"
boundary_of_region_is_subset_of_partition_boundaries_2: shows "\<gamma> \<subseteq> two_chain_boundary two_chain_typeII"
rev_eqvt: fixes pi :: "('b \<times> 'b) list"   and l :: "'a list" shows "pi \<bullet> rev l = rev (pi \<bullet> l)"
assms_2: shows "(True, p, q) \<in> set (np gs bs hs data)"
always_hyp_change_not_Lim: fixes s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "\<And>x. s (f \<triangleright> Suc x) \<noteq> s (f \<triangleright> x)" shows "\<not> learn_lim \<psi> {f} s"
P_inf_def: fixes s :: "'s"   and P :: "'s stream \<Rightarrow> bool" shows "P_inf s P = (\<Sqinter>cfg\<in>cfg_on s. emeasure (T cfg) {x \<in> space St. P x})"
cong_add_trans_2: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and n :: "'a"   and y :: "'a" assumes "[a = x + b] (mod n)"   and "[x = y] (mod n)" shows "[a = y + b] (mod n)"
zero_ne_eSuc: fixes n :: "enat" shows "0 \<noteq> eSuc n"
prod_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less_eq zero (f a)" shows "less_eq zero (comm_monoid_mult.prod times one f A)"
alphaAbs_all_equal_or_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "alphaAbs_all_equal_or_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
fimageE: fixes b :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e fset" assumes "b |\<in>| f |`| A" obtains x :: "'e" where "b = f x" and "x |\<in>| A"
ring_isomorphism_def: fixes \<eta> :: "'d \<Rightarrow> 'e"   and R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and R' :: "'e set"   and addition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multiplication' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero' :: "'e"   and unit' :: "'e" shows "ring_isomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit' \<equiv> ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit' \<and> bijective_map \<eta> R R'"
map\<^sub>0_def: fixes a :: "'a" shows "F.map\<^sub>0 a \<equiv> src\<^sub>C (F a)"
raw_subst_id: fixes t :: "'a" shows "abs_pred (\<lambda>t. \<forall>env. id_env env \<longrightarrow> subst t env = t) t"
simps_3: fixes p :: "'a poly"   and q :: "'a poly" shows "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
add_canc2: fixes x :: "'a"   and y :: "'a" assumes "x \<preceq>\<^sub>R y" shows "x \<oplus> rquot y x = y"
is_discrete: fixes Obj :: "'a set"   and f :: "('a, 'a) concrete_category.arr" shows "partial_magma.arr (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x)) f = partial_magma.ide (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x)) f"
left_gpv_parametric': fixes A :: "'m \<Rightarrow> 'p \<Rightarrow> bool"   and C :: "'n \<Rightarrow> 'q \<Rightarrow> bool"   and R :: "'o \<Rightarrow> 'r \<Rightarrow> bool"   and C' :: "'s \<Rightarrow> 'u \<Rightarrow> bool"   and R' :: "'t \<Rightarrow> 'v \<Rightarrow> bool" shows "(rel_gpv'' A C R ===> rel_gpv'' A (rel_sum C C') (rel_sum R R')) left_gpv left_gpv"
stable_ordIso1: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "stable r"   and "r' =o r" shows "stable r'"
Collect_minus_eq: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "{x. P x} - {x. Q x} = {x. P x \<and> \<not> Q x}"
induct: fixes P :: "'c common_primitive match_expr \<Rightarrow> bool"   and a0 :: "'c common_primitive match_expr" assumes "\<And>uu_. P (Match (Src (IpAddrNetmask uu_ 0)))"   and "\<And>uv_. P (Match (Dst (IpAddrNetmask uv_ 0)))"   and "\<And>iface. P (Match (IIface iface))"   and "\<And>iface. P (Match (OIface iface))"   and "P (Match (Prot ProtoAny))"   and "\<And>m c. P (Match (L4_Flags (TCP_Flags m c)))"   and "\<And>ctstate. P (Match (CT_State ctstate))"   and "\<And>va. P (Match (Src (IpAddr va)))"   and "\<And>va vc. P (Match (Src (IpAddrNetmask va (Suc vc))))"   and "\<And>va vb. P (Match (Src (IpAddrRange va vb)))"   and "\<And>va. P (Match (Dst (IpAddr va)))"   and "\<And>va vc. P (Match (Dst (IpAddrNetmask va (Suc vc))))"   and "\<And>va vb. P (Match (Dst (IpAddrRange va vb)))"   and "\<And>va. P (Match (Prot (Proto va)))"   and "\<And>v. P (Match (Src_Ports v))"   and "\<And>v. P (Match (Dst_Ports v))"   and "\<And>v. P (Match (MultiportPorts v))"   and "\<And>v. P (Match (Extra v))"   and "\<And>m. P m \<Longrightarrow> P (MatchNot m)"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)"   and "P MatchAny" shows "P a0"
fps_subdegree_mult_fps_X_1: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "subdegree (fps_X * f) = subdegree f + 1"
chain: shows "\<forall>i. (f i, f (Suc i)) \<in> R"
are_perspective_from_line_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points" shows "are_perspective_from_line A B C A' B' C' \<equiv> distinct6 A B C A' B' C' \<longrightarrow> triangle A B C \<longrightarrow> triangle A' B' C' \<longrightarrow> line A B \<noteq> line A' B' \<longrightarrow> line A C \<noteq> line A' C' \<longrightarrow> line B C \<noteq> line B' C' \<longrightarrow> col (Pascal_Property.inter (line A B) (line A' B')) (Pascal_Property.inter (line A C) (line A' C')) (Pascal_Property.inter (line B C) (line B' C'))"
dom_closed: fixes f :: "'a" assumes "right \<omega> f" shows "right \<omega> (local.dom f)"
purgeIdle_getGMUserCom_purge: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list"   and sl' :: "('a, 'b, 'c) state list"   and GH :: "'b set" assumes "GM_sec_model.wffp do sl"   and "GM_sec_model.wffp do sl'"   and "list_all2 (GM_sec_model.f GH) sl sl'" shows "GM_sec_model.purge GH (map getGMUserCom (GM_sec_model.purgeIdle (tl sl))) = GM_sec_model.purge GH (map getGMUserCom (GM_sec_model.purgeIdle (tl sl')))"
Tree_wf_def: shows "Tree_wf \<equiv> {(xa, x). Tree_wfp xa x}"
QR_lemma_03: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.f p q = QR.m p q"
LeadsTo_def: fixes A :: "'d set"   and B :: "'d set" shows "A \<longmapsto>w B \<equiv> {F. F \<in> reachable F \<inter> A \<longmapsto> B}"
least_eq: shows "(LEAST n. Hermite_of_row_i ass res A' i $ i $ n \<noteq> (0::'b)) = (LEAST n. A' $ i $ n \<noteq> (0::'b))"
mem_Collect_eq: fixes a :: "'ag"   and P :: "'ag \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
incidence_system_axioms: shows "incidence_system \<V>' \<B>'"
gstep_eq_conv: fixes c :: "(cmd, local_state, global_state) Gen_Scheduler.global_config option"   and c' :: "(cmd, local_state, global_state) Gen_Scheduler.global_config option" shows "((c, c') \<in> li'.s1.gstep) = (\<exists>cc. c = Some cc \<and> c' \<in> li'.s1.gstep_succ cc)"
cut_less_absorb: fixes I :: "'b set"   and t :: "'b" shows "I \<down>< t \<down>< t = I \<down>< t"
d'_1: fixes x :: "'a" assumes "x \<in> {a--b}" shows "\<forall>y\<in>cball x (d' x). Bx x \<le> f y \<bullet> rot (a - b)"
length_f: fixes xs :: "bool list"   and \<eta> :: "nat"   and ys :: "bool list" assumes "length xs = len \<eta>"   and "length ys = len \<eta>" shows "length (f \<eta> xs ys) = len \<eta>"
trs_def: fixes r :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "trs r = {. inpt r .} \<circ> [: r :]"
hull_mon: fixes A :: "'b list set"   and B :: "'b list set" assumes "A \<subseteq> B" shows "\<langle>A\<rangle> \<subseteq> \<langle>B\<rangle>"
set_list_R1_eq: fixes A :: "'d set"   and B :: "'e set" shows "{x. setr x \<bind> fsts \<subseteq> A \<and> setr x \<bind> snds \<subseteq> B} = {x. setl x \<subseteq> UNIV \<and> setr x \<subseteq> {x. fsts x \<subseteq> A \<and> snds x \<subseteq> B}}"
iter_def: fixes x :: "'a" shows "iter x \<equiv> \<Sqinter> range ((^) x)"
wf_measure: fixes f :: "'k \<Rightarrow> nat" shows "wf (measure f)"
find_graph_occurence_edges: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and E :: "('e \<times> 'g \<times> 'g) set"   and f :: "('g \<times> 'f) set" assumes "chain S"   and "finite E"   and "univalent f"   and "on_triple f `` E \<subseteq> edges (chain_sup S)" shows "\<exists>i. on_triple f `` E \<subseteq> edges (S i)"
carrier_vs_is_self: fixes N :: "'c set" shows "carrier (V.vs N) = N"
fw'_spec: fixes m :: "nat \<times> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat" shows "fw' m n k \<le> SPEC (\<lambda>M'. M' = uncurry (fw (curry m) n k))"
indets_map_indets: fixes f :: "'b \<Rightarrow> 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "inj_on f (indets p)" shows "indets (map_indets f p) = f ` indets p"
sublist_appendI: fixes xs :: "'a list"   and ps :: "'a list"   and ss :: "'a list" shows "sublist xs (ps @ xs @ ss)"
establish_rwof_invar: fixes m0 :: "'c nres"   and I :: "'c \<Rightarrow> bool"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and s :: "'c" assumes "m0 \<le>\<^sub>n SPEC I"   and "\<And>s. \<lbrakk>rwof m0 cond step s; I s; cond s\<rbrakk> \<Longrightarrow> step s \<le>\<^sub>n SPEC I"   and "rwof m0 cond step s" shows "I s"
simps_3: fixes As :: "'a attree list"   and s :: "'a set \<times> 'a set" shows "(\<turnstile>As \<oplus>\<^sub>\<or>\<^bsup>s\<^esup>) = (case As of [] \<Rightarrow> fst s \<subseteq> snd s | [a] \<Rightarrow> \<turnstile>a \<and> fst s \<subseteq> fst (attack a) \<and> snd (attack a) \<subseteq> snd s | a # ab # list \<Rightarrow> \<turnstile>a \<and> fst (attack a) \<subseteq> fst s \<and> snd (attack a) \<subseteq> snd s \<and> \<turnstile>ab # list \<oplus>\<^sub>\<or>\<^bsup>(fst s - fst (attack a), snd s)\<^esup>)"
False: shows "\<not> (c = 0 \<or> c' = 0)"
cl5: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set"   and T :: "('a \<times> 'a set) set" shows "R \<cdot> S \<cdot> (T \<cdot> {}) = R \<cdot> (S \<cdot> (T \<cdot> {}))"
case_duality: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - x \<sqinter> - y \<squnion> - x \<sqinter> - z = (- x \<squnion> - y) \<sqinter> (- - x \<squnion> - z)"
ereal_of_enat_ge_zero_cancel_iff: fixes n :: "enat" shows "(0 \<le> ereal_of_enat n) = (0 \<le> n)"
mult_sum_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" shows "(\<otimes>\<^bsub>sum_group I G\<^esub>) = (\<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i)"
bounds_2: fixes n :: "nat" shows "(\<Sum>j = 0..<n. 1 / real (fact (0 + j)) * real_of_float x ^ j) \<le> real_of_float (ub_exp_horner prec n (((\<lambda>i. i + 1) ^^ 0) 1) (fact 0) x)"
eval_trm_fvi_cong: fixes t :: "'b trm"   and v :: "'b list"   and v' :: "'b list" assumes "\<forall>x\<in>fv_trm t. v ! x = v' ! x" shows "eval_trm v t = eval_trm v' t"
fractrel_iff_normalize_quot_eq: fixes x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" shows "fractrel x y = (normalize_quot x = normalize_quot y \<and> snd x \<noteq> (0::'b) \<and> snd y \<noteq> (0::'b))"
iso_tuple_snd_def: fixes isom :: "('a, 'c, 'b) tuple_isomorphism" shows "iso_tuple_snd isom = snd \<circ> repr isom"
downset_set_def: fixes X :: "'b set" shows "\<Down> X = {y. \<exists>x\<in>X. y \<le> x}"
ord_strict_p_monomial_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c" shows "ord_strict_p p (monomial c v) = (c \<noteq> (0::'c) \<and> (p = 0 \<or> lt_of_nat_term_order cmp_term (lt p) v))"
drinks2_vend_insufficient2: fixes r :: "nat \<Rightarrow>f value option"   and x1 :: "int" assumes "r $ 2 = Some (Num x1)"   and "x1 < 100" shows "possible_steps drinks2 2 r STR ''vend'' [] = {|(2, vend_fail)|}"
foo28a: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators" assumes "x \<in> dom (C a)" shows "dom (C a) \<noteq> {}"
interleave_cons1: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b" assumes "l \<in> a \<otimes> b" shows "x # l \<in> x # a \<otimes> b"
h1: shows "i mod 2 = 0 \<or> odd i \<and> i mod 2 = 1"
Sup_UNIV: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<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" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup UNIV = top"