hcju/isabelleps huggingface dataset:
l11_21_a: fixes B :: "'p"   and A :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "B Out A C"   and "A B C CongA A' B' C'" shows "B' Out A' C'"
ASSUME_\<Theta>_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and f\<^sub>0 :: "'d \<Rightarrow> 'e"   and s\<^sub>0 :: "'d"   and R :: "('f \<times> 'e) set"   and \<Theta> :: "(((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'f) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "ASSUME_\<Theta> \<pi> f\<^sub>0 s\<^sub>0 R \<Theta> \<equiv> HT'set_r (\<lambda>f' s'. (f' s', f\<^sub>0 s\<^sub>0) \<in> R) \<pi> \<Theta>"
simps_1: fixes t :: "'b" shows "release_all None t = None"
holfb_i: shows "(\<lambda>z. inverse (f z - a)) holomorphic_on ball z r - {z}"
inf_right_idem: fixes x :: "'a"   and y :: "'a" shows "inf (inf x y) y = inf x y"
succ_min_lvar_not_in_bounds_2: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and xr :: "nat" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "xr \<in> lvars (\<T> s)"   and "xr \<in> rvars (\<T> s')" shows "\<forall>x\<in>lvars (\<T> s). x < xr \<longrightarrow> in_bounds x \<langle>\<V> s\<rangle> (\<B> s)"
set_joinR: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.set_tree (joinR l x r) = Tree2.set_tree l \<union> {x} \<union> Tree2.set_tree r"
star_rtc_least_eq: fixes x :: "'a"   and y :: "'a" assumes "(1::'a) + x + y \<cdot> y = y" shows "x\<^sup>\<star> \<le> y"
coprime_cong_prod_nat: fixes A :: "'a set"   and m :: "'a \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "\<And>i j. \<lbrakk>i \<in> A; j \<in> A; i \<noteq> j\<rbrakk> \<Longrightarrow> coprime (m i) (m j)"   and "\<And>i. i \<in> A \<Longrightarrow> [x = y] (mod m i)" shows "[x = y] (mod prod m A)"
payload_parts_singleton: fixes X :: "msg"   and Y :: "msg" assumes "X \<in> parts {Y}"   and "Y \<in> payload" shows "X \<in> payload"
simps_2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and delta :: "'f"   and inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs option" shows "qOp delta inp binp #[[rho]] = qOp delta (lift (qPsubst rho) inp) (lift (qPsubstAbs rho) binp)"
rule_gen: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" assumes "\<And>\<alpha>. [\<phi> \<alpha> in v]" shows "[\<^bold>\<forall>\<alpha>. \<phi> \<alpha> in v]"
tn''_def: shows "tn'' = [t1, t2, t3]"
carrier_polynomial_shell: fixes K :: "'a set"   and p :: "'a list" assumes "subring K (L\<lparr>carrier := A\<rparr>)"   and "p \<in> carrier (K [X]\<^bsub>L\<lparr>carrier := A\<rparr>\<^esub>)" shows "p \<in> carrier (poly_ring (L\<lparr>carrier := A\<rparr>))"
rel_funE: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'c"   and y :: "'d" assumes "(A ===> B) f g"   and "A x y" obtains "B (f x) (g y)"
fimageE: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b fset" assumes "b |\<in>| f |`| A" obtains x :: "'b" where "b = f x" and "x |\<in>| A"
g_def: shows "g = (\<lambda>z. (1 - z) / (1 - (1 - x * y) * z))"
f_Exec_N_eq_f_Exec_Stream_nth: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n \<le> length xs" shows "f_Exec_Comp_N trans_fun n xs c = (c # f_Exec_Comp_Stream trans_fun xs c) ! n"
Equality_ax_def: shows "Equality_ax \<equiv> {\<lbrakk>\<guillemotleft>refl_ax\<guillemotright>\<rbrakk>e0, \<lbrakk>\<guillemotleft>eq_cong_ax\<guillemotright>\<rbrakk>e0, \<lbrakk>\<guillemotleft>mem_cong_ax\<guillemotright>\<rbrakk>e0, \<lbrakk>\<guillemotleft>eats_cong_ax\<guillemotright>\<rbrakk>e0}"
lcm_0_iff_int: fixes m :: "int"   and n :: "int" shows "(lcm m n = 0) = (m = 0 \<or> n = 0)"
ext: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. f x = g x" shows "f = g"
splitFace_edges_g'_vs: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face"   and f12 :: "face"   and f21 :: "face"   and g' :: "graph" assumes "pre_splitFace g ram1 ram2 f []"   and "(f12, f21, g') = splitFace g ram1 ram2 f []" shows "\<E> g' = \<E> g \<union> {(ram1, ram2), (ram2, ram1)}"
pwd_spy_id_pubk_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_id_pubk n)"
islimpt_approachable_le: fixes x :: "'a"   and S :: "'a set" shows "x islimpt S = (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x \<le> e)"
last_in_set: fixes as :: "'ap list" assumes "as \<noteq> []" shows "last as \<in> set as"
epair_eqD: fixes e :: "'a set"   and x :: "'a"   and y :: "'a" assumes "epair e = Some (x, y)" shows "x \<noteq> y \<and> e = {x, y}"
abs_eq: fixes x :: "nat set" shows "monom_of_set x = Abs_poly_mapping (\<lambda>x. if finite x \<and> x \<in> x then 1 else 0)"
distinct_3: fixes x1 :: "'a" shows "prot_atom.Atom x1 \<noteq> SetType"
set_integral_Un: fixes A :: "'d set"   and B :: "'d set"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "A \<inter> B = {}"   and "set_integrable M A f"   and "set_integrable M B f" shows "set_lebesgue_integral M (A \<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f"
that: shows "eq_\<I>_gpv A \<I> gpv gpv'"
order_pderiv2: fixes p :: "'b poly"   and a :: "'b"   and n :: "nat" assumes "pderiv p \<noteq> 0"   and "order a p \<noteq> 0" shows "(order a (pderiv p) = n) = (order a p = Suc n)"
select_convs_1: fixes V :: "'a set"   and N :: "'a set"   and C :: "'a set"   and more :: "'b" shows "V \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = V"
Group_Ugp: fixes G :: "('c, 'd) Group_scheme" assumes "Ugp G" shows "Group G"
i: shows "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> i"
extend_1: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> sub.diff_fun_space" shows "extend f \<in> diff_fun_space"
conjugation_prop2: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and z :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "g z \<cdot> y \<le> g (z \<cdot> f y)"
z_facts_7: shows "z \<notin> Fvars \<tau>"
is_type_class: fixes P :: "'a prog"   and C :: "String.literal" shows "is_type P (Class C) = is_class P C"
assms_2: shows "ipresIGAbs h hA MOD MOD'"
LIMSEQ_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and k :: "nat" assumes "f \<longlonglongrightarrow> a" shows "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
simps_5: fixes Inv :: "'a"   and b :: "bexp"   and c :: "'a acom"   and P :: "'a"   and Inv' :: "'a"   and b' :: "bexp"   and c' :: "'a acom"   and P' :: "'a" shows "({Inv} WHILE b DO c {P} \<sqsubseteq> {Inv'} WHILE b' DO c' {P'}) = (b = b' \<and> c \<sqsubseteq> c' \<and> Inv \<sqsubseteq> Inv' \<and> P \<sqsubseteq> P')"
ts'_def: shows "check \<C> (e # es) ts = ts'"
Par2F: fixes Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> P \<parallel> Q'"
f2: shows "mix_pmf \<alpha> p q \<approx>[\<R>] mix_pmf \<alpha> (mix_pmf Up B W) q"
some_NF_def: fixes r :: "('b \<times> 'b) set"   and x :: "'b" shows "some_NF r x = (SOME y. (x, y) \<in> r\<^sup>* \<and> y \<in> NF r)"
le_add_diff: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "c \<subseteq># b + c - a"
H: fixes m :: "nat"   and Ax :: "int mpoly"   and A' :: "int mpoly" shows "m < n \<longrightarrow> A \<noteq> Ax * MPoly_Type.monom (monomial (Suc 0) x') 1 + A' \<or> x' \<in> vars A' \<or> MPoly_Type.degree Ax x' \<noteq> m"
nba_g_def: fixes A :: "('d, 'c) nba" shows "nba_g A \<equiv> \<lparr>g_V = UNIV, g_E = E_of_succ (successors A), g_V0 = initial A\<rparr>"
S2P_Collect: fixes \<phi> :: "'c \<Rightarrow> bool" shows "S2P (Collect \<phi>) = \<phi>"
subcls_is_class1: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "P \<turnstile> C \<preceq>\<^sup>* D"   and "is_class P D" shows "is_class P C"
simps_4: fixes v :: "String.literal"   and va :: "typ"   and uu :: "nat"   and uv :: "Core.term" shows "subst_bv2 (Ct v va) uu uv = Ct v va"
is_category: fixes C1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "product_category C1 C2" shows "category (product_category.comp C1 C2)"
update_convs_6: fixes isin' :: "(room \<Rightarrow> guest set) \<Rightarrow> room \<Rightarrow> guest set"   and owns :: "room \<Rightarrow> guest option"   and currk :: "room \<Rightarrow> key"   and issued :: "key set"   and cards :: "guest \<Rightarrow> (key \<times> key) set"   and roomk :: "room \<Rightarrow> key"   and isin :: "room \<Rightarrow> guest set"   and safe :: "room \<Rightarrow> bool"   and more :: "'a" shows "isin_update isin' \<lparr>owns = owns, currk = currk, issued = issued, cards = cards, roomk = roomk, isin = isin, safe = safe, \<dots> = more\<rparr> = \<lparr>owns = owns, currk = currk, issued = issued, cards = cards, roomk = roomk, isin = isin' isin, safe = safe, \<dots> = more\<rparr>"
tendsto_ennreal_iff: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and x :: "real" assumes "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 \<le> x" shows "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F = (f \<longlongrightarrow> x) F"
rt_strictly_fresher_eqI: fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "i \<in> kD rt1"   and "i \<in> kD rt2"   and "nsqn rt1 i = nsqn rt2 i"   and "\<pi>\<^sub>5 (the (rt2 i)) < \<pi>\<^sub>5 (the (rt1 i))" shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
akra_bazzi_term_ceiling_subtract: fixes b :: "real"   and x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and c :: "real" assumes "0 < b"   and "b < 1"   and "real x\<^sub>0 \<le> b * real x\<^sub>1 - c"   and "1 \<le> c + (1 - b) * real x\<^sub>1" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 b (\<lambda>x. nat \<lceil>b * real x - c\<rceil>)"
principal_decomposition: fixes k :: "nat" assumes "principal_dchar c \<noteq> principal_dchar c" shows "principal_dchar c k = \<Phi>.principal.primitive_extension k * principal_dchar c k"
eval_until_res: fixes I :: "\<I>"   and nt :: "nat"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and res :: "event_data option list set list"   and auxlist' :: "(nat \<times> event_data option list set \<times> event_data option list set) list" assumes "eval_until I nt auxlist = (res, auxlist')" shows "res = map proj_thd (takeWhile (check_before I nt) auxlist)"
word_bw_comms_1: fixes x :: "'b word"   and y :: "'b word" shows "x AND y = y AND x"
mod_minus_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (uminus (modulo a b)) b = modulo (uminus a) b"
h_g: fixes y :: "nat"   and n :: "nat" assumes "y \<le> n + 1" shows "g' n y = (\<Sum>x\<le>n + 1. h n x y)"
abs_eq: fixes xa :: "'a \<Rightarrow>\<^sub>0 'b"   and x :: "'a \<Rightarrow>\<^sub>0 'b" shows "lex_pp_strict (PP xa) (PP x) = lex_pm_strict xa x"
\<dd>_foldr_Plus: fixes xs :: "'b language list"   and s :: "'b language"   and a :: "'b" shows "\<dd> (foldr Plus xs s) a = foldr Plus (map (\<lambda>r. \<dd> r a) xs) (\<dd> s a)"
infzeros: fixes y :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "\<forall>x<y. a * x\<^sup>2 + b * x + c = 0" shows "a = 0 \<and> b = 0 \<and> c = 0"
ir_post: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare P c c' Q"   and "\<And>s s'. Q' s s' \<Longrightarrow> Q s s'" shows "ir_hoare P c c' Q'"
f1: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real" assumes "Table0 f1 f2 f1' f2' e c" shows "0 < f1"
HasFoo_accessible: fixes P :: "pname" shows "tprg \<turnstile> Iface HasFoo accessible_in P"
c_def: fixes k :: "nat" shows "c k = 1 / (1 + (4 * real k + 2 * real n)) - x\<^sup>2 / (3 + (4 * real k + 2 * real n))"
constmap_def: fixes A :: "'c set"   and B :: "'d set" shows "constmap A B = (\<lambda>x\<in>A. SOME y. y \<in> B)"
none: assumes "fmlookup m x = None"   and "fmlookup n x = None" shows "thesis"
LFP_lowerbound: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "x \<in> carrier L"   and "f x \<sqsubseteq>\<^bsub>L\<^esub> x" shows "LFP\<^bsub>L\<^esub> f \<sqsubseteq>\<^bsub>L\<^esub> x"
x_def: shows "x = (SOME x. x \<in> r / s)"
connected_componentI: fixes T :: "'a set"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "connected T"   and "T \<subseteq> S"   and "x \<in> T"   and "y \<in> T" shows "connected_component S x y"
compatAtm_def: fixes atm :: "atom" shows "Example_PL.compatAtm atm \<equiv> \<forall>s t. Example_PL.indisAbbrev s t \<longrightarrow> Example_PL.indisAbbrev (aval atm s) (aval atm t)"
take_induct: fixes P :: "'e\<cdot>('f, 'g) errorT \<Rightarrow> bool"   and x :: "'e\<cdot>('f, 'g) errorT" assumes "\<And>n. P (errorT_take n\<cdot>x)" shows "P x"
\<mu>LTL_implies_valid_prefix: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<in> \<mu>LTL"   and "w \<Turnstile>\<^sub>n \<phi>" shows "\<exists>i. af \<phi> (prefix i w) \<sim>\<^sub>C true\<^sub>n"
wf_syscls_def: fixes P :: "'b prog" shows "wf_syscls P \<equiv> (\<forall>C\<in>{Object, Throwable, Thread}. is_class P C) \<and> (\<forall>C\<in>sys_xcpts. P \<turnstile> C \<preceq>\<^sup>* Throwable)"
inPREDD: fixes v :: "nat"   and u :: "nat" assumes "PRED v = Some u" shows "v \<in> N \<or> (\<exists>i\<le>d. v \<in> Vd i)"
mask_eq_iff_w2p: fixes n :: "nat"   and w :: "'b word" assumes "n < size w" shows "(w AND mask n = w) = (w < 2 ^ n)"
small_fpairs: fixes r :: "V" shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
mset_subset_eq_add_mset_cancel: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(a, A \<subseteq># a, B) = (A \<subseteq># B)"
sel_2: fixes xs :: "'d list" shows "stl (cycle xs) = cycle (tl xs @ [hd xs])"
coeff_pochhammer_poly: fixes n :: "nat"   and k :: "nat" shows "coeff (pochhammer_poly n) k = of_nat (stirling n k)"
basis_inI: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "indep_in \<E> X"   and "\<And>x. x \<in> \<E> - X \<Longrightarrow> \<not> indep_in \<E> (insert x X)" shows "basis_in \<E> X"
OrderingSetMap_axioms_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "OrderingSetMap_axioms less_eq less_eq' P f \<equiv> \<forall>a b. a \<in> P \<longrightarrow> b \<in> P \<longrightarrow> less_eq a b \<longrightarrow> less_eq' (f a) (f b)"
remdups_adj_Nil_iff: fixes xs :: "'a list" shows "(remdups_adj xs = []) = (xs = [])"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
h1: shows "distinct (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))"
gen_lossless_gpv_parametric: shows "rel_fun (=) (rel_fun (rel_\<I> (BNF_Def.Grp UNIV id) (=)) (rel_fun (rel_gpv (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV id)) (=))) gen_lossless_gpv gen_lossless_gpv"
assms: shows "\<forall>E F G. \<lbrakk>E, F, G\<rbrakk> p"
NF_I: fixes a :: "'b"   and A :: "('b \<times> 'b) set" assumes "\<And>b. (a, b) \<notin> A" shows "a \<in> NF A"
less_eq_finite_4_def: fixes x :: "finite_4"   and y :: "finite_4" shows "(x \<le> y) = (case (x, y) of (a\<^sub>1, x) \<Rightarrow> True | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True | (a\<^sub>2, _) \<Rightarrow> False | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True | (a\<^sub>3, _) \<Rightarrow> False | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, _) \<Rightarrow> False)"
diff_eq: fixes x :: "'c"   and y :: "'c" shows "x - y = x \<sqinter> - y"
simps: fixes P :: "'a prog"   and a1 :: "ty"   and a2 :: "ty" shows "P \<turnstile> a1 \<le> a2 = ((\<exists>T. a1 = T \<and> a2 = T) \<or> (\<exists>C D. a1 = Class C \<and> a2 = Class D \<and> P \<turnstile> C \<preceq>\<^sup>* D) \<or> (\<exists>C. a1 = NT \<and> a2 = Class C))"
PO_rhoare_defs_1: fixes pre :: "('a \<times> 'b) set"   and Ra :: "('a \<times> 'a) set"   and Rc :: "('b \<times> 'b) set"   and post :: "('a \<times> 'b) set" shows "{pre} Ra, Rc {> post} \<equiv> pre O Rc \<subseteq> Ra O post"
induct: fixes x :: "'a"   and \<delta> :: "('a, 'b) ta_rule set"   and Q0 :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> f_accessible_alt \<delta> Q0"   and "\<And>q0. q0 \<in> Q0 \<Longrightarrow> P q0"   and "\<And>q l qs q'. \<lbrakk>q \<in> f_accessible_alt \<delta> Q0; P q; q \<rightarrow> l qs \<in> \<delta>; q' \<in> set qs\<rbrakk> \<Longrightarrow> P q'" shows "P x"
more_arith_simps_8: fixes a :: "'b"   and b :: "'b" shows "a * - b = - (a * b)"
unitary_gen_iff: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary_gen M = (\<exists>a b k. k \<noteq> 0 \<and> mat_det (a, b, - cnj b, cnj a) \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m (a, b, - cnj b, cnj a))"
iT_Div_iWeakUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<W> t2 I \<oslash> k. Q t2) = (P (t1 div k). t1 \<W> t2 I. Q (t2 div k))"
assms_2: fixes dir :: "('i, 'a) Direction"   and x :: "nat"   and c :: "'a" assumes "dir = Positive \<or> dir = Negative"   and "a = LE dir x c"   and "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"   and "\<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)" shows "P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (set_unsat [i, LI dir s x] (update\<B>\<I> (UBI_upd dir) i x c s))"
frrestriction_vsubset_rel: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>r\<^sub>\<bullet> A \<subseteq>\<^sub>\<circ> r"
split_of_bool_asm: fixes P :: "'a \<Rightarrow> bool"   and p :: "bool" shows "P (of_bool p) = (\<not> (p \<and> \<not> P (1::'a) \<or> \<not> p \<and> \<not> P (0::'a)))"
sat_models_def: fixes \<phi> :: "'a ltln\<^sub>P" shows "sat_models \<phi> = {\<A>. \<A> \<Turnstile>\<^sub>P rep_ltln\<^sub>P \<phi>}"
pcs_def: fixes xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list" shows "pcs xt \<equiv> \<Union>(f, t, C, h, d)\<in>set xt. {f..<t}"
x: shows "row x A \<in> {row i A |i. row i A \<noteq> 0}"
init_fin_simulation_Wakeup2: fixes t :: "'t"   and sx1 :: "status \<times> 'x1"   and m1 :: "'m1"   and sx2 :: "status \<times> 'x2"   and m2 :: "'m2"   and ta2 :: "('l, 't, status \<times> 'x2, 'm2, 'w, 'o action) thread_action"   and sx2' :: "status \<times> 'x2"   and m2' :: "'m2" assumes "t \<turnstile> (sx1, m1) \<approx>i (sx2, m2)"   and "sx1 \<approx>iw sx2"   and "r2.init_fin t (sx2, m2) ta2 (sx2', m2')"   and "Notified \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub> \<or> WokenUp \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub>" shows "\<exists>ta1 sx1' m1'. r1.init_fin t (sx1, m1) ta1 (sx1', m1') \<and> t \<turnstile> (sx1', m1') \<approx>i (sx2', m2') \<and> ta_bisim init_fin_bisim ta1 ta2"
SWF_def: fixes swf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and Pcond :: "'d set \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> bool" shows "SWF swf A Is Pcond \<equiv> \<forall>P. Pcond A Is P \<longrightarrow> rpr A (swf P)"
element_ptr_casts_commute: fixes node_ptr :: "(_) node_ptr"   and element_ptr :: "(_) element_ptr" shows "(cast node_ptr = Some element_ptr) = (cast element_ptr = node_ptr)"
SCp_swap_applies_3: fixes \<Gamma> :: "'a formula multiset"   and G :: "'a formula"   and H :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> G, H, \<Delta>" shows "\<Gamma> \<Rightarrow> H, G, \<Delta>"
Exit_Def_empty: fixes V :: "'c" assumes "V \<in> Def Exit" shows "False"
subgraph_slim: shows "subgraph (with_proj slim) (with_proj G)"
psi_odd_pos: fixes n :: "nat" shows "0 \<le> psi_odd n"
inject_6: fixes x61 :: "exp"   and x62 :: "exp"   and x63 :: "exp"   and y61 :: "exp"   and y62 :: "exp"   and y63 :: "exp" shows "(EIf x61 x62 x63 = EIf y61 y62 y63) = (x61 = y61 \<and> x62 = y62 \<and> x63 = y63)"
distinct_disc_13: fixes event :: "('d, 'e, 'f) event" assumes "isSend event" shows "\<not> isRecvMarker event"
closure_subset: fixes S :: "'d set" shows "S \<subseteq> closure S"
tempName_def: fixes c :: "cmd" shows "tempName c \<equiv> replicate (Suc (maxVnameLen c)) CHR ''a''"
firstPosEqualZero: fixes a :: "'b"   and m :: "'b"   and M' :: "'b list" shows "(firstPos a (m # M') = 0) = (a = m)"
N2_enabled_at_g: shows "\<turnstile> pc2 = #g \<longrightarrow> Enabled (<N2>_(x, y, sem, pc1, pc2))"
distinct_9: fixes x6 :: "'b ptrm" shows "PUnit \<noteq> PFst x6"
exec_stop2: fixes \<sigma> :: "'b control_state_scheme" assumes "return_status \<sigma>" shows "exec_stop \<sigma>"
ltl_const_equiv_equivp: shows "equivp (\<sim>\<^sub>C)"
embed_bothWays_Field_bij_betw: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'d \<Rightarrow> 'c" assumes "Well_order r"   and "Well_order r'"   and "embed r r' f"   and "embed r' r f'" shows "bij_betw f (Field r) (Field r')"
FAddSA_dom: fixes S :: "'d"   and A :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<notin> dom A" shows "(A [f+] (S, SA)) = A"
set_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "set (x21 # x22) = Set.insert x21 (set x22)"
fps_cutoff_def: fixes n :: "nat"   and f :: "'b fps" shows "fps_cutoff n f = Abs_fps (\<lambda>i. if i < n then f $ i else (0::'b))"
simps: fixes a1 :: "bval"   and a2 :: "exp" shows "bs_val a1 a2 = ((\<exists>n. a1 = BNat n \<and> a2 = ENat n) \<or> (\<exists>\<rho> \<rho>' x e. a1 = BClos x e \<rho> \<and> a2 = ELam x (psubst ((x, EVar x) # \<rho>') e) \<and> bs_env \<rho> \<rho>' \<and> FV (ELam x (psubst ((x, EVar x) # \<rho>') e)) = {}))"
SN_rel_on_empty: fixes R :: "('a \<times> 'a) set"   and T :: "'a set" shows "SN_rel_on R {} T = SN_on R T"
Bex_set_list_ex: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
ref_in: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<in> lists \<langle>G\<rangle>" shows "Ref G us \<in> lists G\<^sub>+"