hcju/isabelleps huggingface dataset:
enorm_def: shows "enorm \<equiv> \<lambda>x. sqrt (\<Sum>i<n. (x i)\<^sup>2)"
non_interference_induct_case_sub2: fixes n :: "nat"   and s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "(user_seq_exe n s1 \<and> user_seq_exe n s2 \<longrightarrow> (\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2))) \<and> user_seq_exe (Suc n) s1 \<and> user_seq_exe (Suc n) s2" shows "\<exists>t1. Some t1 = (case SEQ n s1 of None \<Rightarrow> None | Some x \<Rightarrow> NEXT x) \<and> (\<exists>t2. Some t2 = (case SEQ n s2 of None \<Rightarrow> None | Some x \<Rightarrow> NEXT x) \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2)"
Suc_in_imp_not_subset_iMODb: fixes n :: "nat"   and S :: "nat set"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "n \<in> S"   and "Suc n \<in> S"   and "m \<noteq> Suc 0" shows "\<not> S \<subseteq> [ r, mod m, c ]"
hash_blindable_simps_2: fixes h :: "'b \<Rightarrow> 'a"   and y :: "'a blindable\<^sub>h" shows "hash_blindable h (Blinded y) = y"
arr_iff_in_hom: fixes f :: "'a" shows "C.arr f = \<guillemotleft>f : C.dom f \<Rightarrow>\<^sub>C C.cod f\<guillemotright>"
fref_weaken_pre: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'd"   and R :: "('b \<times> 'a) set"   and S :: "('c \<times> 'd) set" assumes "\<And>x. P x \<longrightarrow> P' x"   and "(f, h) \<in> [P']\<^sub>f R \<rightarrow> S" shows "(f, h) \<in> [P]\<^sub>f R \<rightarrow> S"
comp_apply: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and x :: "'g" shows "(f \<circ> g) x = f (g x)"
one_poly_eq_simps_1: shows "(1 = [:1::'b:]) = True"
g_isEmpty_def: fixes ops :: "('a, 'b, 'c, 'd) map_basic_ops_scheme"   and m :: "'c" shows "StdBasicMapDefs.g_isEmpty ops m \<equiv> StdBasicMapDefs.g_size_abort ops 1 m = 0"
abs_uexpr_def: fixes u :: "('a, 'b) uexpr" shows "\<bar>u\<bar> = uop abs u"
elims: fixes x :: "('a, 'b, 'c) unboxed"   and y :: "Unboxed.type option" assumes "typeof x = y" obtains uu_ :: "'a" where "x = OpDyn uu_" and "y = None"   | uv_ :: "'b" where "x = OpUbx1 uv_" and "y = Some Ubx1"   | uw_ :: "'c" where "x = OpUbx2 uw_" and "y = Some Ubx2"
ffmember_filter: fixes x :: "'c"   and P :: "'c \<Rightarrow> bool"   and A :: "'c fset" shows "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)"
finsert_fminus_single: fixes a :: "'c"   and A :: "'c fset" shows "finsert a (A |-| {|a|}) = finsert a A"
finite_member: fixes t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "finite (Collect (RBT_Set2.member t))"
simps_2: fixes x :: "bool"   and xs :: "bool list" shows "rbl_succ (x # xs) = (if x then False # rbl_succ xs else True # xs)"
simps_2: fixes x :: "'b" shows "is_NoMsg (Msg x) = False"
Mp_rel_iI: fixes f1 :: "'i list"   and f2 :: "'a mod_ring poly"   and f3 :: "int poly" assumes "poly_rel f1 f2"   and "MP_Rel f3 f2" shows "Mp_rel_i f1 f3"
direct_decomp_Cons_subsetI: fixes A :: "'a set"   and s :: "'a set"   and ss :: "'a set list" assumes "direct_decomp A (s # ss)"   and "\<And>s0. s0 \<in> set ss \<Longrightarrow> (0::'a) \<in> s0" shows "s \<subseteq> A"
perm_simps_8: fixes p :: "perm"   and var :: "Vars.var"   and exp :: "exp"   and assn :: "assn" shows "p \<bullet> ACons var exp assn = ACons (p \<bullet> var) (p \<bullet> exp) (p \<bullet> assn)"
prems_1: shows "H \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F)"
object_ptr_kinds_eq: shows "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h' \<turnstile> object_ptr_kinds_M|\<^sub>r"
inf_apply: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "inf f g x = inf (f x) (g x)"
sym: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and b :: "'b"   and a :: "'b" assumes "comparator cmp" shows "(cmp b a = Equiv) = (cmp a b = Equiv)"
inj_singleton: fixes A :: "'a set" shows "inj_on (\<lambda>x. {x}) A"
pred_stream_cfg_on: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and xs :: "('s \<times> ('c \<Rightarrow> real) set) cfg stream" assumes "cfg \<in> R_G.valid_cfg"   and "MDP.MC.enabled cfg xs" shows "pred_stream (\<lambda>cfg. state cfg \<in> \<S> \<and> cfg \<in> R_G.cfg_on (state cfg)) xs"
left_plus_below_circ: fixes x :: "'a" shows "x \<sqinter> top \<le> top"
times_hmultiset_monoL: fixes a :: "hmultiset"   and b :: "hmultiset"   and c :: "hmultiset" assumes "a < b"   and "0 < c" shows "a * c < b * c"
DN_simps_1: fixes \<mu> :: "('a E.term, 'a) arr" assumes "arr \<mu>" shows "B.arr (DN \<mu>)"
UN_insert: fixes B :: "'b \<Rightarrow> 'a set"   and a :: "'b"   and A :: "'b set" shows "\<Union> (B ` insert a A) = B a \<union> \<Union> (B ` A)"
xyroots: shows "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
valid_red: fixes n :: "nat" assumes "3 \<le> n" shows "valid (replicate n R)"
l_inv: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
obtains_1_1_2: fixes c1 :: "nat \<Rightarrow>f value option"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list"   and c2 :: "nat \<Rightarrow>f value option" assumes "obtains 1 c1 drinks2 1 r t"   and "obtains 1 c2 drinks 1 r t" shows "c1 = r \<and> c2 = r"
poly_mapping_size_def: fixes f :: "'c \<Rightarrow> nat"   and g :: "'d \<Rightarrow> nat"   and m :: "'c \<Rightarrow>\<^sub>0 'd" shows "poly_mapping_size f g m = g (0::'d) + (\<Sum>k\<in>keys m. Suc (f k + g (lookup m k)))"
assms_6: fixes p' :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and q' :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "(p', q') \<in> snd ` set ps'"   and "fst p' \<noteq> 0"   and "fst q' \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"
prems_3: fixes ys :: "'b list"   and y :: "'b" shows "P \<epsilon> (ys \<cdot> [y])"
elims_3: fixes x :: "int \<times> int"   and xa :: "int \<times> int" assumes "\<not> le_rat_raw x xa" obtains a :: "int" and b :: "int" and c :: "int" and d :: "int" where "x = (a, b)" and "xa = (c, d)" and "\<not> a * d * (b * d) \<le> c * b * (b * d)"
offset_poly_pCons: fixes a :: "'a"   and p :: "'a poly"   and h :: "'a" shows "offset_poly (pCons a p) h = smult h (offset_poly p h) + pCons a (offset_poly p h)"
measurable_envelopeD_3: fixes M :: "'b measure"   and A :: "'b set"   and E :: "'b set"   and F :: "'b set" assumes "measurable_envelope M A E"   and "F \<in> sets M" shows "emeasure M (F \<inter> E) = outer_measure_of M (F \<inter> A)"
disjoint_family_on_disjoint_image: fixes A :: "'b \<Rightarrow> 'c set"   and I :: "'b set" assumes "disjoint_family_on A I" shows "disjoint (A ` I)"
ipassmt_disjoint: shows "ipassmt_sanity_disjoint ipassmt"
vdomain_vrange_is_vempty: shows "(\<D>\<^sub>\<circ> r = 0) = (\<R>\<^sub>\<circ> r = 0)"
edges_part: fixes u :: "'a"   and a :: "('b, 'c) edgeAnnot"   and v :: "'a" assumes "(u, a, v) \<in> edges fg" shows "proc_of fg u = proc_of fg v"
interior_of_maximal_eq: fixes X :: "'c topology"   and T :: "'c set"   and S :: "'c set" assumes "openin X T" shows "(T \<subseteq> X interior_of S) = (T \<subseteq> S)"
induct_list012: fixes P :: "'g list \<Rightarrow> bool"   and xs :: "'g list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y zs. \<lbrakk>P zs; P (y # zs)\<rbrakk> \<Longrightarrow> P (x # y # zs)" shows "P xs"
inversion_id_iff: fixes a :: "complex_homo" shows "(a = inversion a) = (a \<noteq> \<infinity>\<^sub>h \<and> to_complex a * cnj (to_complex a) = 1)"
foldl_Cons: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'e"   and x :: "'f"   and xs :: "'f list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
sets_in_Pi_aux: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and F :: "'d \<Rightarrow> 'e set" assumes "finite I"   and "\<And>j. j \<in> I \<Longrightarrow> {x \<in> space (M j). x \<in> F j} \<in> sets (M j)" shows "{x \<in> space (Pi\<^sub>M I M). x \<in> Pi I F} \<in> sets (Pi\<^sub>M I M)"
Domain_unfold: fixes r :: "('e \<times> 'f) set" shows "Domain r = {x. \<exists>y. (x, y) \<in> r}"
tarski_first3_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "tarski_first3 C \<equiv> (\<forall>a b. C a b b a) \<and> (\<forall>a b p q r s. C a b p q \<and> C a b r s \<longrightarrow> C p q r s) \<and> (\<forall>a b c. C a b c c \<longrightarrow> a = b)"
lcf_monom_1: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "monom (UP Zp) a n (deg Zp (monom (UP Zp) a n)) = a"
simps_60: fixes x7 :: "('a, 'b, 'c) expr"   and x61 :: "('a, 'b, 'c) expr"   and x62 :: "('a, 'b, 'c) expr" shows "Rfork x7 \<noteq> Assign x61 x62"
isotone_def: fixes f :: "'a \<Rightarrow> 'a" shows "ord.isotone (\<lambda>x y. y \<le> x) f \<equiv> \<forall>x y. y \<le> x \<longrightarrow> f y \<le> f x"
parent_lower_bound: fixes l :: "int"   and c :: "int" shows "(l \<le> parent l c) = (l < c)"
NEUT_def: fixes t :: "lam" shows "NEUT t \<equiv> (\<exists>a. t = Var a) \<or> (\<exists>t1 t2. t = App t1 t2)"
proj_decomp: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" assumes "FGModuleEnd G smult V T"   and "\<And>g. g \<in> V \<Longrightarrow> T (T g) = T g" shows "V = (ker T \<inter> V) \<oplus> T ` V"
mem_UnitsI: fixes u :: "'a" assumes "additive.invertible u"   and "u \<in> R" shows "u \<in> additive.Units"
CDPlayer_CTRL_Init_def: shows "CDPlayer_CTRL_Init = ''CDEmpty''"
get_host_is_l_get_host: shows "l_get_host get_host get_host_locs"
\<delta>: shows "ball \<xi> \<delta> \<subseteq> S"
STAR_singleton: fixes x :: "'a" shows "*s* {x} = {star_of x}"
ends: fixes t :: "real" assumes "t \<in> {0..1}" shows "pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
igFreshIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igFreshIGOp MOD \<equiv> \<forall>ys y delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igFreshInp MOD ys y inp \<and> igFreshBinp MOD ys y binp \<longrightarrow> igFresh MOD ys y (igOp MOD delta inp binp)"
Red_I_of_Red_F_subset: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and N' :: "'a set"   and N :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "N' \<subseteq> Red_F N" shows "Red_I N \<subseteq> Red_I (N - N')"
inducts: fixes x1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and x2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and P :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool" assumes "consistent x1 x2"   and "\<And>ae a as \<Gamma> e S. \<lbrakk>a_consistent (ae, a, as) (\<Gamma>, e, S); \<And>x. x \<in> thunks \<Gamma> \<Longrightarrow> ae x = up\<cdot>0\<rbrakk> \<Longrightarrow> P (ae, a, as) (\<Gamma>, e, S)" shows "P x1 x2"
assms_2: fixes R :: "pi"   and S :: "pi" assumes "(R, S) \<in> Rel" shows "R \<leadsto><Rel> S"
c4: shows "(\<exists>l\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni l)) x) = (\<exists>(a', b', c')\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni (a', b', c'))) x)"
Range_Un_eq: fixes A :: "('b \<times> 'a) set"   and B :: "('b \<times> 'a) set" shows "Range (A \<union> B) = Range A \<union> Range B"
arev_arev_raw: fixes a :: "'b" assumes "a \<in> arcs H" shows "rev_H' (rev_H' a) = a"
dw_closed_UNIV: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "relation.dw_closed r UNIV"
eq_iff: fixes a :: "'b list"   and b :: "'b list" shows "(a = b) = (??.Sublist.prefix b a \<and> ??.Sublist.prefix a b)"
arg_inv_2kpi: fixes z :: "complex" assumes "z \<noteq> 0" shows "\<exists>x. arg (1 / z) = - arg z + 2 * real_of_int x * pi"
less_def: fixes s :: "'a"   and s' :: "'a" shows "s \<prec> s' \<equiv> s \<preceq> s' \<and> s \<noteq> s'"
f: fixes e :: "'v \<times> 'v" shows "(\<lambda>i. f_i i e) \<longlonglongrightarrow> ennreal (f' e)"
Q3'_def: shows "Q3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q3"
Refl: fixes X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "(X, X) \<in> qSwapped"
compact_translation: fixes s :: "'b set"   and a :: "'b" assumes "compact s" shows "compact ((+) a ` s)"
Quotient_double: shows "Quotient (\<lambda>x y. IEEE.is_nan x \<and> IEEE.is_nan y \<or> x = y) abs_double rep_double cr_double"
MAX_SET_ELIM': fixes P :: "'c set"   and R :: "'c \<Rightarrow> bool" assumes "finite P"   and "P \<noteq> {}"   and "\<forall>x. (\<forall>y. y \<in> P \<longrightarrow> y \<le> x) \<and> x \<in> P \<longrightarrow> R x" shows "R (Max P)"
lookup_default_def: fixes d :: "'b"   and m :: "('c, 'b) mapping"   and k :: "'c" shows "Mapping.lookup_default d m k = (case Mapping.lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
rev_flow_image_eq: fixes x :: "'a"   and S :: "real set" shows "flow0 x ` S = rev.flow0 x ` uminus ` S"
exhaust: fixes y :: "secLevel" obtains "y = Low"   | "y = High"
the_singleton_def: shows "the_singleton \<equiv> THE id. \<forall>k id'. \<parallel>id'\<parallel>\<^bsub>k\<^esub> \<longrightarrow> id' = id"
minus_eq_not_plus_1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "uminus a = plus (not a) one"
check: shows "fd2 = Fundef (if \<Sigma>2 = [None] then zs else ys) ar"
sel_2: fixes x1 :: "'a list"   and x2 :: "'a" shows "concl_of (Infer x1 x2) = x2"
assms_2: shows "Deliver (i1, Add i2 e) \<in> set xs"
simps_2: fixes prog :: "cmd"   and c :: "cmd" shows "labels_nodes prog (_Entry_) c = False"
properfactorI2: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a divides\<^bsub>G\<^esub> b"   and "\<not> a \<sim>\<^bsub>G\<^esub> b" shows "properfactor G a b"
slength_sappend: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" shows "slength\<cdot>(xs :@ ys) = slength\<cdot>xs + slength\<cdot>ys"
simps_4: fixes Ts :: "typ list"   and T :: "typ"   and body :: "Core.term" shows "typ_of1 Ts (Abs T body) = Option.bind (typ_of1 (T # Ts) body) (\<lambda>x. Some (T \<rightarrow> x))"
int_pts_def: fixes p :: "nat list" shows "int_pts p = set (butlast p)"
\<E>_RF: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" shows "\<E>\<^bsub>\<Gamma>\<^esub> (RF\<^bsub>\<Gamma>\<^esub> S) = \<E>\<^bsub>\<Gamma>\<^esub> S"
a: shows "p \<cdot> q \<cdot> x1 \<cdot> q \<cdot> (q \<cdot> r1 \<cdot> y1 + !q \<cdot> r2 \<cdot> y2)\<^sup>\<dagger> \<cdot> (q \<cdot> !r1 + !q \<cdot> !r2) = p \<cdot> q \<cdot> x1 \<cdot> q \<cdot> (r1 \<cdot> y1)\<^sup>\<dagger> \<cdot> !r1"
join_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x + z \<le> y + z"
prec_enh: shows "\<forall>ppred p q f g x y. np - maxfaults \<le> count ppred np \<and> okRead1 f y ppred \<and> okRead1 g y ppred \<and> okRead2 f g x ppred \<and> ppred p \<and> ppred q \<longrightarrow> \<bar>cfn p f - cfn q g\<bar> \<le> \<pi> x y"
bit_take_bit_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (take_bit m a) n = (n < m \<and> bit a n)"
ldrop_enat: fixes n :: "nat"   and xs :: "'b llist" shows "ldrop (enat n) xs = ldropn n xs"
fresh_2: fixes a :: "name"   and x1 :: "name"   and x2 :: "('a, 'b, 'c) input" shows "a \<sharp> \<nu>x1x2 = a \<sharp> [x1].x2"
vec_matrix: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d" shows "vec_to_iarray (A $ i) = matrix_to_iarray A !! mod_type_class.to_nat i"
out_var_def: fixes x :: "'a \<Longrightarrow> 'c" shows "$x\<acute> = x ;\<^sub>L snd\<^sub>L"
completeness_game_def: fixes h :: "'a"   and w :: "nat"   and e :: "nat" shows "Schnorr_\<Sigma>.completeness_game h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (check h a e z)))"
seq_bot: fixes c :: "'a::refinement_lattice" shows "\<bottom> ; (c::'a::refinement_lattice) = \<bottom>"
fls_regpart_eq0_imp_nonpos_subdegree: fixes f :: "'a fls" assumes "fls_regpart f = 0" shows "fls_subdegree f \<le> 0"
WT_callee: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "\<I>' \<turnstile>g callee s x \<surd>"
dbl_inc_simps_5: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit1 k)"
result: shows "closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = None"
element_of_subseqs_then_subset: fixes l :: "'c list"   and l' :: "'c list" assumes "l \<in> set (subseqs l')" shows "set l \<subseteq> set l'"
field_differentiable_within_tan: fixes z :: "'a"   and s :: "'a set" assumes "cos z \<noteq> (0::'a)" shows "tan field_differentiable at z within s"
K_I: shows "K = C \<circ> I \<circ> C"
exec_meth_append: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ins' :: "'addr instr list" assumes "exec_meth ci P ins xt t h s ta h' s'" shows "exec_meth ci P (ins @ ins') xt t h s ta h' s'"
edgesUnsimp: shows "edges G \<union> set (filter_compliant_stateful_ACS G M edgesList) \<inter> set (filter_IFS_no_violations G M edgesList) = edges G"
p_def: fixes s :: "'s"   and t :: "'s"   and n :: "nat" shows "p s t n = T.prob s {\<omega> \<in> space (T s). (s ## \<omega>) !! n = t}"
confluence_modulo_equivalence: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>1' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 \<leadsto>\<^sup>* s\<^sub>2"   and "s\<^sub>1' \<leadsto>\<^sup>* s\<^sub>2'"   and "s\<^sub>1 \<approx> s\<^sub>1'"   and "reachable s\<^sub>1"   and "infinite UNIV"   and "infinite UNIV" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> s\<^sub>2 \<leadsto>\<^sup>* s\<^sub>3 \<and> s\<^sub>2' \<leadsto>\<^sup>* s\<^sub>3'"
ab: shows "(a, b) = splitAt (hd vs) (between vs ram2 ram1)"
that: fixes \<psi> :: "'a \<Rightarrow> real" assumes "diff_fun k charts \<psi>"   and "\<And>x. x \<in> carrier \<Longrightarrow> 0 \<le> \<psi> x"   and "\<And>x. x \<in> carrier \<Longrightarrow> \<psi> x \<le> 1"   and "\<And>x. x \<in> A \<Longrightarrow> \<psi> x = 1"   and "csupport_on carrier \<psi> \<inter> carrier \<subseteq> U" shows "thesis"
neg_odd: fixes n :: "int" assumes "n < 0" shows "odd (int_option_enumeration (Some n))"
RH'': shows "relH {a. a < lim (the_state \<sigma>') \<and> a \<notin> new_addrs h as (the_state \<sigma>')} (the_state \<sigma>') (the_state \<sigma>'')"
inject_1: fixes x11 :: "String.literal"   and x12 :: "typ list"   and y11 :: "String.literal"   and y12 :: "typ list" shows "(Ty x11 x12 = Ty y11 y12) = (x11 = y11 \<and> x12 = y12)"
that: fixes u :: "'t"   and v :: "'t" assumes "u \<in> keys p"   and "v \<in> keys q"   and "component_of_term u = component_of_term v"   and "w = term_of_pair (pp_of_term u + pp_of_term v, component_of_term u)" shows "thesis"
ladder_E_L': shows "LeftDerivationLadder \<alpha> E L' \<beta>"
head_polybound0: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "polybound0 (head p)"
imbal_ins_Unbal: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree"   and tw :: "bool"   and y :: "'a"   and s :: "'a tree" assumes "ins n d x t = Unbal t'" shows "real (imbal (node tw t' y s)) - real (imbal (node tw t y s)) \<le> 1"
lookup_default: fixes x :: "'c"   and d :: "'b"   and m :: "('c, 'b) mapping" shows "Mapping.lookup (Mapping.default x d m) x = Some (Mapping.lookup_default d m x)"
abc_Hoare_plus_unhalt2': fixes Q :: "nat list \<Rightarrow> bool"   and B :: "abc_inst list"   and P :: "nat list \<Rightarrow> bool"   and A :: "abc_inst list"   and args :: "nat list"   and n :: "nat" assumes "{Q} B \<up>"   and "{P} A {Q}"   and "A \<noteq> []"   and "P args" shows "abc_notfinal (abc_steps_l (0, args) (A [+] B) n) (A [+] B)"
read_buffer_RelS: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" shows "(\<langle>A.read_buffer, mds\<^sub>A_of mds\<^sub>0, mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>C.read_buffer\<^sub>C, mds\<^sub>0, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
induct: fixes x :: "fm"   and P :: "fm \<Rightarrow> bool" assumes "x \<in> induction_axioms"   and "\<And>j i A. atom j \<sharp> (i, A) \<Longrightarrow> P (A(i::=Zero) IMP SyntaxN.All i (SyntaxN.All j (A IMP A(i::=Var j) IMP A(i::=Eats (Var i) (Var j)))) IMP SyntaxN.All i A)" shows "P x"
path_connectedin_path_image: fixes X :: "'a topology"   and g :: "real \<Rightarrow> 'a" assumes "pathin X g" shows "path_connectedin X (g ` {0..1})"
eta_conversion_lemma_0: fixes v :: "i"   and p :: "\<o>" shows "[\<^bold>\<lambda>\<^sup>0 p \<^bold>= p in v]"
scale_right_distrib: fixes a :: "'a"   and x :: "'c"   and y :: "'c" shows "s2 a (x + y) = s2 a x + s2 a y"
flist_arb_inv: fixes A :: "'b fset" shows "finset (flist_arb A) = A"
big_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> L F g1"   and "f2 \<in> L F g2" shows "(\<lambda>x. f1 x * f2 x) \<in> L F (\<lambda>x. g1 x * g2 x)"