hcju/isabelleps huggingface dataset:
red_preserves_sconf: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"   and "P,E,hp s \<turnstile> e : T"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
smooth_on_compose: fixes k :: "enat"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "k-smooth_on T f"   and "k-smooth_on S g"   and "open S"   and "open T"   and "g ` S \<subseteq> T" shows "k-smooth_on S (f \<circ> g)"
assms_2: fixes a :: "'b" assumes "a \<in> A" shows "f a = (0::'a)"
wordinterval_eq_comm: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_eq r1 r2 = wordinterval_eq r2 r1"
no_trailing: fixes xs' :: "'a list" assumes "list_all2 R xs xs'" shows "no_trailing ((=) (0::'a)) xs' = no_trailing ((=) (arith_ops_record.zero ops)) xs"
simps_1: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and uu :: "'b" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2" shows "WBT.delete \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 uu \<langle>\<rangle> = \<langle>\<rangle>"
cls_def: fixes cf :: "CAttId" shows "cls cf = (case cf of CounterImpl'CounterImpl'value \<Rightarrow> CounterImpl | CounterImpl'Counter'dummy \<Rightarrow> CounterImpl | _ \<Rightarrow> UndoCounter)"
PPf_def: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf" shows "Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf \<equiv> \<lambda>t1 t2. Generic_Syntax.psubst var Var FvarsT Fvars subst Pf [(t1, Generic_Syntax.Variable var 11), (t2, Generic_Syntax.Variable var 10)]"
agree_plus2: fixes \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<nu>' :: "(real, 'c) vec \<times> (real, 'c) vec"   and t1 :: "('d, 'c) trm"   and t2 :: "('d, 'c) trm" assumes "Vagree \<nu> \<nu>' (FVDiff (Plus t1 t2))" shows "Vagree \<nu> \<nu>' (FVDiff t2)"
restriction_clm_res_disjoint: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "restrict v (res ts) c \<sqinter> restrict v (clm ts) c = bot"
elims_2: fixes x :: "'c \<Rightarrow> nat"   and xa :: "nat \<times> nat list \<times> 'c list" assumes "count_inv x xa" obtains f :: "'c \<Rightarrow> nat" and u :: "nat" and ns :: "nat list" and xs :: "'c list" where "x = f" and "xa = (u, ns, xs)" and "\<forall>x. count (mset xs) x = f x"
Osum_minus_Id: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Total r"   and "Total r'"   and "\<not> r \<subseteq> Id"   and "\<not> r' \<subseteq> Id" shows "(r \<union>o r') - Id \<subseteq> r - Id \<union>o r' - Id"
in_not_in_iff2: fixes n :: "nat"   and i :: "nat_int" shows "(n \<^bold>\<notin> i) = (\<not> n \<^bold>\<in> i)"
induct: fixes P :: "nat list \<Rightarrow> bool"   and a0 :: "nat list" assumes "P []"   and "\<And>n ns. P ns \<Longrightarrow> P (Suc (Suc (Suc (Suc n))) # ns)"   and "\<And>ns. P ns \<Longrightarrow> P (0 # ns)"   and "\<And>ns. P ns \<Longrightarrow> P (Suc 0 # ns)"   and "\<And>ns. P ns \<Longrightarrow> P (Suc (Suc 0) # ns)"   and "\<And>ns. P ns \<Longrightarrow> P (Suc (Suc (Suc 0)) # ns)" shows "P a0"
st_def: shows "st = (\<lambda>x. SOME r. x \<in> HFinite \<and> r \<in> \<real> \<and> r \<approx> x)"
pgen_lossless_gpv_parametric': shows "((=) ===> rel_\<I> C (=) ===> rel_gpv'' A C (=) ===> (=)) pgen_lossless_gpv pgen_lossless_gpv"
sep_add_left_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj a b"   and "sep_disj b c"   and "sep_disj a c" shows "plus b (plus a c) = plus a (plus b c)"
u_def: shows "u = (\<lambda>n. v (r (Suc n)) - v (r n))"
indexed_zero_def: shows "indexed_const \<zero> = (\<lambda>_. \<zero>)"
add_le_imp_le_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less_eq (plus c a) (plus c b)" shows "less_eq a b"
is_maxsplitter_def: fixes P :: "'a list \<Rightarrow> bool"   and f :: "'a list \<Rightarrow> 'a list \<times> 'a list" shows "is_maxsplitter P f = (\<forall>xs ps qs. (f xs = (ps, qs)) = (xs = ps @ qs \<and> is_maxpref P ps xs))"
Const\<^sub>0_def: fixes c :: "'b" shows "Const\<^sub>0 c \<equiv> monomial c 0"
simps_13: fixes x4 :: "ty"   and x2 :: "qtname" shows "ArrayT x4 \<noteq> IfaceT x2"
swap_simps_8: fixes pi :: "(vrs \<times> vrs) list"   and x :: "tyvrs" shows "pi \<bullet> x = x"
unique_connecting_path_set_2: fixes v :: "nat"   and w :: "nat" assumes "v \<in> V"   and "w \<in> V" shows "w \<in> set (v \<leadsto> w)"
wffp_NE: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do sl" shows "sl \<noteq> []"
span_mem: fixes E :: "rat vec set"   and u :: "rat vec" assumes "E \<subseteq> Rn"   and "u \<in> E" shows "u \<in> gs.span E"
mcont_fun_lub_Sup: fixes M :: "('b \<Rightarrow> 'a) set" assumes "Complete_Partial_Order.chain (fun_ord (\<le>)) M"   and "\<forall>f\<in>M. mcont \<Or> (\<sqsubseteq>) Sup (\<le>) f" shows "mcont \<Or> (\<sqsubseteq>) Sup (\<le>) (fun_lub Sup M)"
CNV_eqD: fixes x :: "'a"   and y :: "'a" assumes "CNV x y" shows "x = y"
subset_trans: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "A \<subseteq> C"
bres_sol: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> (x \<rightarrow> y) = y) = (\<exists>z. x \<cdot> z = y)"
totient_primorial': fixes k :: "nat" shows "real (totient (primorial' k)) = real (primorial' k) * (\<Prod>i<k. 1 - 1 / real (nth_prime i))"
compare_MkI_MkI: fixes x :: "int"   and y :: "int" shows "compare\<cdot>(MkI\<cdot>x)\<cdot>(MkI\<cdot>y) = (if x < y then LT else if y < x then GT else EQ)"
Fourier_sum_offset: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<Sum>k\<le>2 * n. Fourier_coefficient f k * trigonometric_set k t) = (\<Sum>k\<le>2 * n. Fourier_coefficient (\<lambda>x. f (x + t)) k * trigonometric_set k 0)"
connected_component_in_connected_components_of: fixes X :: "'b topology"   and a :: "'b" shows "(connected_component_of_set X a \<in> connected_components_of X) = (a \<in> topspace X)"
hyp: shows "fib (gcd n (m mod n)) = gcd (fib n) (fib (m mod n))"
resr_antitonel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<le> y \<longrightarrow> y \<rightarrow> z \<le> x \<rightarrow> z"
Nat_def: shows "HOLZF.Nat \<equiv> Sep HOLZF.Inf (\<lambda>N. \<exists>n. nat2Nat n = N)"
inter_coset_fold: fixes A :: "'a set"   and xs :: "'a list" shows "A \<inter> List.coset xs = fold Set.remove xs A"
list_all2_opt_True: fixes n :: "nat"   and A_zs :: "nat set list"   and A_x :: "nat set"   and A_xs :: "nat set list"   and A_y :: "nat set"   and A_ys :: "nat set list"   and A_neg :: "nat set list"   and zs :: "'b option list set list"   and x :: "'b option list set"   and xs :: "'b option list set list"   and y :: "'b option list set"   and ys :: "'b option list set list"   and L_neg :: "'b option list set list" assumes "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs @ A_y # A_ys) @ A_neg) ((zs @ x # xs @ y # ys) @ L_neg)"   and "length A_xs = length xs"   and "length A_ys = length ys"   and "length A_zs = length zs" shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ (A_x \<union> A_y) # A_xs @ A_ys) @ A_neg) ((zs @ join x True y # xs @ ys) @ L_neg)"
prv_imp_eqv_transi: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (imp \<phi> \<chi>1)"   and "prv (eqv \<chi>1 \<chi>2)" shows "prv (imp \<phi> \<chi>2)"
dvd_add_left_iff: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c" shows "(a dvd b + c) = (a dvd b)"
fcompare_double_Lt_asym: fixes x :: "double"   and y :: "double" assumes "fcompare_double x y = ccode.Lt"   and "fcompare_double y x = ccode.Lt" shows "False"
rot_circle_top_left_edges_neq: shows "rot_circle_top_edge \<noteq> rot_circle_left_edge"
vcomp_atI: fixes r :: "V"   and s :: "V"   and a :: "V"   and b :: "V"   and c :: "V" assumes "vsv r"   and "vsv s"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s"   and "s\<lparr>b\<rparr> = c"   and "r\<lparr>a\<rparr> = b" shows "(s \<circ>\<^sub>\<circ> r)\<lparr>a\<rparr> = c"
zeta_minus_pole_eq: fixes s :: "complex" assumes "s \<noteq> 1" shows "zeta s - 1 / (s - 1) = pre_zeta 1 s"
dupe1_1: shows "A * D1 + B * H1 =m U1"
set_insert: fixes x :: "'g"   and xs :: "'g list" shows "set (List.insert x xs) = insert x (set xs)"
elims: fixes x :: "atomUni"   and y :: "atomUni fmUni" assumes "substNegInfinityUni x = y" obtains p :: "real \<times> real \<times> real" where "x = EqUni pa__" and "y = allZero' pa__"   | p :: "real \<times> real \<times> real" where "x = LessUni pb__" and "y = alternateNegInfinity' pb__"   | p :: "real \<times> real \<times> real" where "x = LeqUni pc__" and "y = OrUni (alternateNegInfinity' pc__) (allZero' pc__)"   | p :: "real \<times> real \<times> real" where "x = NeqUni p" and "y = negUni (allZero' p)"
am_add_op_def: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_left_monoid antidomain_op one times" shows "antidomain_left_monoid.am_add_op antidomain_op times x y \<equiv> antidomain_op (times (antidomain_op x) (antidomain_op y))"
h_4: fixes h' :: "real" assumes "h' \<in> {0<--<h}" shows "(flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> B"
count_ne_remove: fixes x :: "'a"   and t :: "'a"   and S :: "'a multiset" assumes "x \<noteq> t" shows "count S x = count (S - {#t#}) x"
star_simulation_right: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z * x \<le> y * z" shows "z * x\<^sup>\<star> \<le> y\<^sup>\<star> * z"
cexpr_sem_dist_dens_cexpr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and dst :: "pdf_dist"   and e2 :: "cexpr"   and V :: "nat set"   and \<sigma> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : dist_param_type dst"   and "\<Gamma> \<turnstile>\<^sub>c e2 : dist_result_type dst"   and "free_vars e1 \<subseteq> V"   and "free_vars e2 \<subseteq> V"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "ennreal (extract_real (cexpr_sem \<sigma> (dist_dens_cexpr dst e1 e2))) = dist_dens dst (cexpr_sem \<sigma> e1) (cexpr_sem \<sigma> e2)"
RMWWrite: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O>s :: "nat set list"   and i :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool" assumes "cond (\<theta>(t \<mapsto> m a))"   and "a \<in> \<O> \<or> a \<in> dom \<S>"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j \<union> dom (\<R>s ! j)"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = {}"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> (\<O>s ! j \<union> dom (\<R>s ! j)) = {}"   and "a \<notin> read_only \<S>" shows "\<O>s,\<R>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
reconstruct_with_type_conversion: fixes f :: "int poly" shows "Polynomial.smult (lead_coeff (of_int_poly f)) (\<Prod>a\<leftarrow>complex_roots_int f. [:- a, 1:]) = of_int_poly f"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T1 :: "'fsym trm"   and T2 :: "'fsym trm" shows "I.satA \<xi> (Eq T1 T2) = (I.int \<xi> T1 = I.int \<xi> T2)"
leof_trans_nofail: fixes a :: "'b nres"   and b :: "'b nres"   and c :: "'b nres" assumes "a \<le>\<^sub>n b"   and "nofail b"   and "b \<le>\<^sub>n c" shows "a \<le>\<^sub>n c"
homeomorphic_space_imp_isomorphic_homology_groups: fixes X :: "'a topology"   and Y :: "'b topology"   and p :: "int" assumes "X homeomorphic_space Y" shows "homology_group p X \<cong> homology_group p Y"
isFault_def: fixes S :: "('a, 'b) xstate" shows "isFault S = (\<exists>f. S = Fault f)"
sub_num_simps_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit1 k) num.One = numeral.numeral one plus (num.Bit0 k)"
seq_implies_weakly_seq: fixes P :: "'a option process" assumes "sequential P" shows "weakly_sequential P"
IH: fixes V :: "'a set" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t (V \<union> W) S"
open_poly_mapping_def: fixes U :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "open U \<equiv> \<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U"
eitherMarkedOrNotMarked: fixes e :: "'b"   and M :: "('b \<times> bool) list" assumes "e \<in> set (elements M)" shows "(e, True) \<in> set M \<or> (e, False) \<in> set M"
Suffixes_altdef: fixes A :: "'b list set" shows "Suffixes A = (\<Union>w\<in>A. set (suffixes w))"
simps_1: fixes ops :: "('c, 'd) approximate_set_ops_scheme" shows "concat_appr ops [] = []"
id: fixes i :: "nat" shows "([],  \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>) \<in> Ax"
mset_single_iff: fixes xs :: "'b list"   and x :: "'b" shows "(mset xs = x, {#}) = (xs = [x])"
finite_lderivs_wf: fixes r :: "'b rexp" shows "finite {\<guillemotleft>lderivs w r\<guillemotright> |w. wf_word n w}"
l_get_root_node_wf_axioms_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_root_node_wf_axioms heap_is_wellformed get_root_node type_wf known_ptrs get_ancestors get_parent \<equiv> ((\<forall>h ptr. heap_is_wellformed h \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> ptr |\<in>| object_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_root_node ptr) \<and> (\<forall>h ptr. h \<turnstile> ok get_root_node ptr \<longrightarrow> ptr |\<in>| object_ptr_kinds h) \<and> (\<forall>h ptr root. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root \<longrightarrow> root |\<in>| object_ptr_kinds h)) \<and> (\<forall>h ptr ancestors ptr' ptr'' root_ptr. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> get_ancestors ptr \<rightarrow>\<^sub>r ancestors \<longrightarrow> ptr' \<in> set ancestors \<longrightarrow> ptr'' \<in> set ancestors \<longrightarrow> h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root_ptr = h \<turnstile> get_root_node ptr'' \<rightarrow>\<^sub>r root_ptr) \<and> (\<forall>h ptr child. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<longrightarrow> h \<turnstile> get_parent child \<rightarrow>\<^sub>r None) \<and> (\<forall>h child ptr root. h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr \<longrightarrow> h \<turnstile> get_root_node (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r root = h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root)"
is_category_with_terminal_object: shows "category_with_terminal_object (\<cdot>)"
diamond_bot_left_edge_def: shows "diamond_bot_left_edge = (1, \<lambda>x. (diamond_x (1 / 2 * x), - (diamond_x (1 / 2 * x) + d / 2)))"
regular_closed_sup: fixes x :: "'a"   and y :: "'a" assumes "regular x"   and "regular y" shows "regular (x \<squnion> y)"
YM4_parts_knows_Spy: fixes A :: "agent"   and NB :: "msg"   and Y :: "msg"   and c :: "msg"   and evs :: "event list" assumes "Gets A \<lbrace>NB, Crypt (shrK A) Y, c\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "c \<in> parts (knows Spy evs)"
eq_onp_def: fixes R :: "'b \<Rightarrow> bool" shows "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
u_in_df: shows "restrict0 carrier u \<in> diff_fun_space"
L: shows "ta_lang (ta_prod TA1 TA2) = L1 \<inter> L2"
ivl_less: fixes x :: "'a ivl"   and y :: "'a ivl" shows "(x < y) = (low x < low y \<or> low x = low y \<and> high x < high y)"
a: shows "c \<sharp> M[x\<turnstile>n>y]"
vconverse_def: fixes A :: "V" shows "vconverse A = (\<lambda>r\<in>\<^sub>\<circ>A. set {\<langle>b, a\<rangle> |a b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r})"
simps_3: fixes x :: "nat"   and e :: "cexpr"   and e1 :: "cexpr"   and e2 :: "cexpr" shows "cexpr_subst x e <e1, e2>\<^sub>c = <cexpr_subst x e e1, cexpr_subst x e e2>\<^sub>c"
residue_a_inv_closed: fixes k :: "nat"   and y :: "int" shows "\<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> y \<in> carrier (residue_ring (p ^ k))"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "\<And>x. hom x = (0::'b) \<Longrightarrow> x = (0::'a)" shows "zero_hom_0_axioms hom"
\<oo>_foldr_Times: fixes xs :: "'b language list"   and s :: "'b language" shows "\<oo> (foldr Times xs s) = (\<forall>x\<in>set (s # xs). \<oo> x)"
rep_abs_rsp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and x1 :: "'b"   and x2 :: "'b" assumes "Quotient3 R Abs Rep"   and "R x1 x2" shows "R x1 (Rep (Abs x2))"
sort_all_coeffs_unsorted_poly_rel_with0: fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> fully_unsorted_poly_rel" shows "sort_all_coeffs p \<le> \<Down> unsorted_poly_rel_with0 (RETURN p')"
squarefree_decompose: fixes n :: "nat" shows "\<Prod>(squarefree_part n) * (square_part n)\<^sup>2 = n"
sff_1: shows "square_free_factorization g (c, hs)"
Stable: fixes A :: "state \<times> state \<Rightarrow> bool"   and sigma :: "behavior"   and P :: "state \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>A"   and "\<turnstile> $P \<and> A \<longrightarrow> P$" shows "sigma \<Turnstile> stable P"
igSwapIPresIGWlsSTR_imp_igSwapIPresIGWls: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igSwapIPresIGWlsSTR MOD" shows "igSwapIPresIGWls MOD"
Rf_model: fixes I :: "'a set"   and N :: "'a literal multiset set" assumes "I \<Turnstile>s N - sr.Rf N" shows "I \<Turnstile>s N"
min_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" shows "ord.min less_eq a b = (if less_eq a b then a else b)"
hyps_2: shows "xs \<cdot> [x] = LynFac w"
v_round_def: fixes r :: "nat"   and r_votes :: "process \<Rightarrow> val option"   and r_decisions :: "process \<Rightarrow> val option" shows "v_round r r_votes r_decisions = {(s, s'). r = v_state.next_round s \<and> no_defection s r_votes r \<and> d_guard r_decisions r_votes \<and> s' = s\<lparr>v_state.next_round := Suc r, votes := (votes s)(r := r_votes), v_state.decisions := v_state.decisions s ++ r_decisions\<rparr>}"
\<beta>_conversion: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and T :: "typ"   and t :: "Core.term"   and u :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "wt_term (sig \<Theta>) (Abs T t)"   and "wf_term (sig \<Theta>) u"   and "\<turnstile>\<^sub>\<tau> u : T" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs T t $ u) (subst_bv u t)"
rep_eq: fixes x :: "('a, 'b) IEEE.float" shows "Rep_float (- x) = (case Rep_float x of (s, e, f) \<Rightarrow> (1 - s, e, f))"
independentD: fixes s :: "'b set"   and t :: "'b set"   and u :: "'b \<Rightarrow> 'a"   and v :: "'b" assumes "local.independent s"   and "finite t"   and "t \<subseteq> s"   and "(\<Sum>v\<in>t. u v *s v) = (0::'b)"   and "v \<in> t" shows "u v = (0::'a)"
axioms_5: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "is_tdghm_axioms \<AA> \<BB> \<FF> \<GG> \<NN>"
simps_7: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<rr> \<lbrace>t\<rbrace>"
eval_r_s: fixes s :: "nat \<Rightarrow> nat option"   and x :: "nat" assumes "r1_bc s" shows "s x = eval (r1_bc.r_s s) [x]"
sub: shows "\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF x2)) \<subseteq> \<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF (InContext x1 x2)))"
wiring_comp_converterI: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv1 :: "('a, 'b, 'c, 'd) converter"   and fg :: "('a \<Rightarrow> 'c) \<times> ('d \<Rightarrow> 'b)"   and \<I>'' :: "('e, 'f) \<I>"   and conv2 :: "('c, 'd, 'e, 'f) converter"   and fg' :: "('c \<Rightarrow> 'e) \<times> ('f \<Rightarrow> 'd)" assumes "wiring \<I> \<I>' conv1 fg"   and "wiring \<I>' \<I>'' conv2 fg'" shows "wiring \<I> \<I>'' (conv1 \<odot> conv2) (fg \<circ>\<^sub>w fg')"