hcju/isabelleps huggingface dataset:
error_free_normal: fixes s :: "state" assumes "normal s" shows "error_free s"
simps_1: shows "mset_tree \<langle>\<rangle> = {#}"
fmember_intro: fixes x :: "'b"   and xs :: "'b fset" assumes "x \<in> \<langle>xs\<rangle>\<^sub>f" shows "x |\<in>| xs"
ts_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [Label n es' lfilledk] : t1s _> t2s"
OSou_iff_weak_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(operational_sound (TRel\<^sup>*) \<and> weak_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*) \<and> weak_reduction_simulation (Rel\<inverse>) (STCal Source Target))"
\<nu>'\<nu>: shows "\<guillemotleft>\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu> : local.dom \<rho> \<Rightarrow> local.dom \<rho> \<star> w' \<star> w\<guillemotright>"
intro: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" assumes "df_system.is_run s\<^sub>0 lstep s" shows "run_axioms s\<^sub>0 lstep s"
character_dc2c: fixes \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>" shows "character G (dc2c \<chi>)"
sel_2: fixes x21 :: "'g"   and x22 :: "'i" shows "generat.output (IO x21 x22) = x21"
fmdom'_notI: fixes m :: "('e, 'd) fmap"   and x :: "'e" assumes "fmlookup m x = None" shows "x \<notin> fmdom' m"
simps_11: fixes x31 :: "'b"   and x32 :: "'b"   and x33 :: "'c"   and x21 :: "'b"   and x22 :: "'b" shows "AllowPortFromTo x31 x32 x33 \<noteq> DenyAllFromTo x21 x22"
stdin_world2: shows "eval getLine (exec (println STR ''Hello World! What is your name?'') world) = String.implode ''corny''"
Says_imp_knows_Spy: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" shows "X \<in> knows Spy evs"
distinct_19: fixes x4 :: "32 word"   and x5 :: "'b" shows "Intg x4 \<noteq> Addr x5"
run_asset_iv_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_asset_iv n s))) \<notin> used s"
power_not_zero: fixes a :: "'b"   and n :: "nat" assumes "a \<noteq> (0::'b)" shows "a ^ n \<noteq> (0::'b)"
in_homI: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and a :: "'a"   and b :: "'a" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a"   and "partial_magma.cod C f = b" shows "partial_magma.in_hom C f a b"
cases: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P \<turnstile> a1 \<prec>\<^sub>S a2" obtains C :: "char list" and Bs :: "base list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list" and D :: "char list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(Bs, rest)\<rfloor>" and "Shares D \<in> set Bs"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "ModuleHom smult V smult' T" shows "VectorSpaceHom smult V smult' T"
rotate_rotate: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list" shows "rotate m (rotate n xs) = rotate (m + n) xs"
mat_addcol_def: fixes a :: "'b"   and k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "addcol a k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = j then a * A $$ (i, l) + A $$ (i, j) else A $$ (i, j))"
Exit_source: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and a :: "'a" assumes "CFGExit sourcenode targetnode kind valid_edge Entry Exit"   and "valid_edge a"   and "sourcenode a = Exit" shows "False"
intros_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and s :: "'a"   and S :: "'a set" assumes "class.ab_group_add plus zero minus uminus"   and "s \<in> insert zero S" shows "s \<in> ab_group_add.group_closure zero minus S"
table_of_mapkey_SomeD: fixes C :: "'e"   and t :: "('d \<times> 'c) list"   and k :: "'d"   and D :: "'e"   and x :: "'c" assumes "table_of (map (\<lambda>(k, x). ((k, C), x)) t) (k, D) = Some x" shows "C = D \<and> table_of t k = Some x"
dist_ell2_def: fixes x :: "'c ell2"   and y :: "'c ell2" shows "dist x y = norm (x - y)"
assms_5: shows "VS\<^sub>1.dim S = VS\<^sub>2.dim T"
nchotomy: shows "\<forall>vote. vote = A \<or> vote = B"
inverse_nonzero_iff_nonzero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(inverse a = zero) = (a = zero)"
finite_measure_Diff': fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> events"   and "B \<in> events" shows "prob (A - B) = prob A - prob (A \<inter> B)"
simps: fixes x :: "'a extT"   and y :: "'a extT" shows "x * y = x \<sqinter> y"
pop2_def: fixes xs :: "int list"   and stk :: "nat list"   and v :: "int" shows "pop2 xs stk v \<equiv> monadic_WHILEIT (\<lambda>_. set stk \<subseteq> {0..<length xs}) (\<lambda>x. case x of [] \<Rightarrow> RETURN False | k # stk \<Rightarrow> ASSERT (k < length xs) \<bind> (\<lambda>_. RETURN (v \<le> xs ! k))) mop_list_tl stk"
graph_union_def: fixes G\<^sub>1 :: "('c, 'd) labeled_graph"   and G\<^sub>2 :: "('c, 'd) labeled_graph" shows "graph_union G\<^sub>1 G\<^sub>2 = LG (edges G\<^sub>1 \<union> edges G\<^sub>2) (vertices G\<^sub>1 \<union> vertices G\<^sub>2)"
IH: shows "\<forall>a<2 ^ n. dist (geod n a) (geod n (Suc a)) = dist x0 y0 / 2 ^ n"
access_update_other: fixes path' :: "'a list"   and path :: "'a list"   and opt :: "('b att_scheme \<times> 'c, 'b att_scheme, 'a) env option"   and root :: "('b att_scheme \<times> 'c, 'b att_scheme, 'a) env"   and uid :: "nat"   and perms :: "perm set" assumes "path' \<parallel> path" shows "access (update path' opt root) path uid perms = access root path uid perms"
tanh_real_gt_neg1: fixes x :: "real" shows "- 1 < tanh x"
trie_iteratei_correct: fixes t :: "('c, 'd) Trie.trie" assumes "invar_trie t" shows "set_iterator (Trie_Impl.iteratei t) (trie_reverse_key ` map_to_set (lookup_trie t))"
lunit_char_3: fixes f :: "'a" assumes "Left.ide f" shows "\<exists>!\<mu>. Left.in_hom \<mu> (Left.L f) f \<and> Left.L \<mu> = Left.comp (\<i>[trg f] \<star> f) (local.inv \<a>[trg f, trg f, f])"
max_ge: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "Max x y \<succeq> x"
add_mset_eq_add_mset_ne: fixes a :: "'b"   and b :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" assumes "a \<noteq> b" shows "(add_mset a A = add_mset b B) = (a \<in># B \<and> b \<in># A \<and> A = add_mset b (remove1_mset a B))"
assms_3: fixes i :: "nat" shows "\<bar>snd (pdevs_apply (snd Y) i)\<bar> = \<bar>snd (pdevs_apply (snd X) i)\<bar>"
rel_mono: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Ra :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_option R \<le> rel_option Ra"
reflp_preserved_f: fixes f :: "('a, 'b, 'c) prog" shows "reflp (preserved f)"
S5: shows "mod7 (c_fst (c_pair 6 (c_pair n1 n2))) = 6"
s_prod_subdistl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> y + x \<cdot> z \<le> x \<cdot> (y + z)"
gcd_eq_0_iff: fixes a :: "'a"   and b :: "'a" shows "(normalization_euclidean_semiring_class.gcd a b = (0::'a)) = (a = (0::'a) \<and> b = (0::'a))"
simps_7: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and Y :: "'b ltln set" shows "(\<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2)[Y]\<^sub>\<mu> = (if \<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2 \<in> Y then true\<^sub>n else \<psi>\<^sub>1[Y]\<^sub>\<mu> M\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>)"
holo1: shows "(\<lambda>z. CLBINT u=0..ereal (exp (- 1)). complex_of_real (- ln u) powr z / complex_of_real (1 - u) * complex_of_real (- ln u + u - 1)) holomorphic_on {z. - 2 < Re z}"
trans_order_nest: fixes r :: "('a \<times> 'a) set"   and R :: "((accent \<times> 'a) list \<times> (accent \<times> 'a) list) set" shows "trans (nest r R)"
Inequality: shows "x = \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t"
CCons: fixes y :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "P (y # xs)" shows "P (x # y # xs)"
add_nonpos_neg: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># {#}"   and "b \<subset># {#}" shows "a + b \<subset># {#}"
X1: fixes a :: "'b" assumes "(w, y) = arc_to_ends G a" shows "a \<notin> arcs G"
inf_cancel_left1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "x \<sqinter> a \<sqinter> (- x \<sqinter> b) = \<bottom>"
assms_2: assumes "\<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 (set_unsat [i, LI dir s x] (update\<B>\<I> (UBI_upd dir) i x c s))"
slice2_irs_SDG_path: fixes n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n \<in> sum_SDG_slice2 n'"   and "valid_SDG_node n'" obtains ns :: "'node SDG_node list" where "n irs-ns\<rightarrow>\<^sub>d* n'"
set_mset_mono: fixes A :: "'c multiset"   and B :: "'c multiset" assumes "A \<subseteq># B" shows "set_mset A \<subseteq> set_mset B"
coplanar_perm_18: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet D A B C"
card_image_le: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "card (f ` A) \<le> card A"
fls_compose_fps_def: fixes f :: "'b fls"   and g :: "'b fps" shows "fls_compose_fps f g = (if f = 0 then 0 else if 0 \<le> fls_subdegree f then fps_to_fls (fls_regpart f oo g) else fps_to_fls (fls_base_factor_to_fps f oo g) / fps_to_fls g ^ nat (- fls_subdegree f))"
tl_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun (list_all2 A) (list_all2 A) tl tl"
sum_list_prod_cong: fixes xys :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<forall>(x, y)\<in>set xys. f x y = g x y" shows "(\<Sum>(x, y)\<leftarrow>xys. f x y) = (\<Sum>(x, y)\<leftarrow>xys. g x y)"
domintros_10: fixes v :: "lop"   and va :: "exp0"   and vc :: "exp0" shows "is_value_dom (Log v va vc)"
prime_ideal_proper: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "prime_ideal R I" shows "I \<noteq> carrier R"
unambiguous_routing_def: fixes rtbl :: "('c, 'd) routing_rule_scheme list" shows "unambiguous_routing rtbl \<equiv> \<forall>rt1 rt2 rr ra. rtbl = rt1 @ rr # rt2 \<longrightarrow> ra \<in> set (rt1 @ rt2) \<longrightarrow> routing_match rr = routing_match ra \<longrightarrow> routing_rule_sort_key rr \<noteq> routing_rule_sort_key ra"
le_supI2: fixes x :: "'a multiset"   and b :: "'a multiset"   and a :: "'a multiset" assumes "x \<subseteq># b" shows "x \<subseteq># a \<union># b"
nneg: fixes x :: "'a"   and i :: "nat" assumes "x \<in> A i" shows "0 \<le> f x"
abc': shows "a \<squnion>\<^bsub>f\<^esub> (b \<squnion>\<^bsub>f\<^esub> c) \<in> A"
fmdom_notD: fixes x :: "'a"   and m :: "('a, 'b) fmap" assumes "x |\<notin>| fmdom m" shows "fmlookup m x = None"
beukers_aux_integral_transform3: fixes w :: "real" assumes "w \<in> {0<..<1}" shows "LBINT (x, y):D. P x * (1 - y) ^ n / (1 - (1 - x * y) * w) = LBINT (x, y):D. (1 - y) ^ n * (x * y * w) ^ n * (1 - x) ^ n / (1 - (1 - x * y) * w) ^ (n + 1)"
tensor_eqI: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "dims A = dims B"   and "vec A = vec B" shows "A = B"
simps_12: fixes v :: "'b" shows "le_rexp (Atom v) One = False"
list_all2_Cons2: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and y :: "'h"   and ys :: "'h list" shows "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
map\<^sub>0_def: fixes a :: "'a" shows "map\<^sub>0 a \<equiv> src\<^sub>B (local.map a)"
in_set_zip: fixes p :: "'g \<times> 'h"   and xs :: "'g list"   and ys :: "'h list" shows "(p \<in> set (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)"
cases: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" obtains e :: "'a" and a :: "'b" and r :: "nat" and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = SkewBinomialHeapStruc.Node e a r ts"
orientable_1_implies_4: fixes x :: "'a" assumes "orientable_1 (x \<squnion> (1::'a))" shows "orientable_4 x"
simps_1: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "spp_of_pair (Inl (p, q)) = spair_spp p q"
differing_vars_lists_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i = sifum_security.differing_vars mem\<^sub>1 (fst (mems ! i)) \<union> sifum_security.differing_vars mem\<^sub>2 (snd (mems ! i))"
assms_2: fixes r :: "'d" shows "\<alpha> (fr r) \<le> Fr r"
real_nat_ceiling_ge: fixes x :: "real" shows "x \<le> real (nat \<lceil>x\<rceil>)"
infsetsum_Un_Int: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "f abs_summable_on A \<union> B" shows "infsetsum f (A \<union> B) = infsetsum f A + infsetsum f B - infsetsum f (A \<inter> B)"
arr_char: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "arr f = (f \<noteq> Null \<and> |Dom f| <o \<AA> \<and> |Cod f| <o \<AA> \<and> Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f))"
correctCompositionIn_def: fixes x :: "specID" shows "correctCompositionIn x \<equiv> ins x = \<Union> (ins ` subcomponents x) - loc x \<and> ins x \<inter> \<Union> (out ` subcomponents x) = {}"
that: assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "thesis"
checker_form: fixes a\<^sub>e :: "real"   and a\<^sub>o :: "real"   and s\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>e :: "real"   and v\<^sub>o :: "real" assumes "interpret_form (Conj (Conj (Less (Var (Suc (Suc 0))) (Var (Suc (Suc (Suc 0))))) (Conj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Var (Suc (Suc (Suc (Suc (Suc 0))))))) (Conj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Var (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Conj (Less (Var 0) (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0))))))))) (Less (Var (Suc 0)) (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0))))))))))))) (Disj (Less (Add (Var (Suc (Suc 0))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc 0)))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var 0))))) (Var (Suc (Suc (Suc 0))))) (Conj (LessEqual (Var (Suc (Suc (Suc 0)))) (Add (Var (Suc (Suc 0))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc 0)))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var 0)))))) (Conj (Less (Add (Var (Suc (Suc 0))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc 0)))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var 0))))) (Add (Var (Suc (Suc (Suc 0)))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var (Suc 0))))))) (Disj (LessEqual (Var (Suc 0)) (Var 0)) (Disj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc 0)))))) (Var (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Disj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Add (Var (Suc (Suc (Suc (Suc (Suc 0)))))) (Minus (Mult (Mult (Var 0) (Inverse (Var (Suc 0)))) (Var (Suc (Suc (Suc (Suc (Suc (Suc 0))))))))))) (Less (Mult (Power (Add (Var (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) (Minus (Var (Suc (Suc (Suc (Suc (Suc 0)))))))) 2) (Inverse (Var (Suc (Suc (Suc (Suc 0))))))) (Mult (Add (Var (Suc (Suc (Suc 0)))) (Minus (Var (Suc (Suc 0))))) (Add (Var (Suc 0)) (Minus (Var 0)))))))))))) [a\<^sub>e, a\<^sub>o, s\<^sub>e, s\<^sub>o, 2, v\<^sub>e, v\<^sub>o, 0]" shows "checker s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o"
simps: fixes p :: "perm"   and x :: "'a"   and y :: "'b" shows "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
NullT_leaf_int: fixes C :: "ITypeId" shows "NullT \<le> InterfaceT C"
rep_eq: fixes x :: "('a, ('b, 'c) mapping) mapping"   and xa :: "'c"   and xb :: "'c \<Rightarrow> 'c"   and xc :: "('a \<times> 'b) set" shows "mapping.rep (upd_nested x xa xb xc) = (\<lambda>a. case Mapping.lookup x a of None \<Rightarrow> if a \<in> fst ` xc then Some (upd_set' Mapping.empty xa xb {b. (a, b) \<in> xc}) else None | Some m' \<Rightarrow> Some (upd_set' m' xa xb {b. (a, b) \<in> xc}))"
card_of_ordLess2: fixes B :: "'a set"   and A :: "'b set" assumes "B \<noteq> {}" shows "(\<nexists>f. f ` A = B) = (|A| <o |B|)"
face_cycle_sets_def: shows "G.face_cycle_sets = G.face_cycle_set ` arcs G"
vinsertI1: fixes x :: "V"   and A :: "V" shows "x \<in>\<^sub>\<circ> vinsert x A"
topological_basis_open: fixes "open" :: "'c set \<Rightarrow> bool"   and B :: "'c set set"   and X :: "'c set" assumes "class.topological_space open"   and "topological_space.topological_basis open B"   and "X \<in> B" shows "open X"
distinct_entries: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "distinct (map fst (RBT_Impl.entries t))"
R1_H2_commute: shows "R1 \<circ> H2 = H2 \<circ> R1"
mset_eqD_set_mset: fixes xs :: "'a list"   and A :: "'a multiset" assumes "mset xs = A" shows "set xs = set_mset A"
not_none_1: shows "obs \<sigma> k \<noteq> None"
map_tl: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "map f (tl xs) = tl (map f xs)"
measurable_distr_eq2: fixes Mg' :: "'c measure"   and Mg :: "'e measure"   and Ng :: "'d measure"   and g :: "'e \<Rightarrow> 'd" shows "Mg' \<rightarrow>\<^sub>M distr Mg Ng g = Mg' \<rightarrow>\<^sub>M Ng"
prime_not_unit': fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "ASSUMPTION (normalization_semidom.prime zero times one normalize x)" shows "\<not> dvd.dvd times x one"
log2_lm4: fixes x :: "nat"   and u :: "nat" assumes "c_in x u = 1" shows "x \<le> log2 u"
g_imp_holomorphic_log: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f holomorphic_on S"   and "continuous_on S g"   and "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on S" and "\<And>z. z \<in> S \<Longrightarrow> f z = exp (g z)"
compute_float_round_down: fixes prec :: "nat"   and m :: "int"   and e :: "int" shows "float_round_down prec (Float m e) = (let d = bitlen \<bar>m\<bar> - int prec - 1 in if 0 < d then Float (div_twopow m (nat d)) (e + d) else Float m e)"
to_nat_from_nat: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and j :: "'d"   and k :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep j = k" shows "mod_type.from_nat Abs k = j"
imp_imp_commute: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla"   and C :: "fmla" assumes "H \<turnstile> B IMP A IMP C" shows "H \<turnstile> A IMP B IMP C"
seqs_def: shows "seqs \<equiv> \<lambda>j0 j K. list_of (a j0) # map (list_of \<circ> (\<lambda>i. b (enum K i) (j0, i))) (list_of {..<j})"
psi_scale_2: fixes r :: "real"   and a :: "real \<times> real"   and b :: "real \<times> real" shows "psi (r *\<^sub>R a) 0 (r *\<^sub>R b) = (if 0 < r then psi a 0 b else if r < 0 then psi b 0 a else True)"
Union_Pow_eq: fixes A :: "'a set" shows "\<Union> (Pow A) = A"
proofTree_def2: shows "proofTree = (\<lambda>x. bounded x \<and> founded subs (SATAxiom \<circ> sequent) x)"
csum_absorb1: fixes r2 :: "('d \<times> 'd) set"   and r1 :: "('e \<times> 'e) set" assumes "Cinfinite r2"   and "r1 \<le>o r2" shows "r2 +c r1 =o r2"
assms_5: shows "h \<turnstile> get_elements_by_class_name ptr' idd \<rightarrow>\<^sub>r results"
atMost_def: fixes u :: "'t" shows "ord_term_lin.atMost u \<equiv> {x. x \<preceq>\<^sub>t u}"
ksimplex_replace_2: fixes p :: "nat"   and n :: "nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat" assumes "ksimplex p n s"   and "a \<in> s"   and "n \<noteq> 0"   and "\<forall>j<n. \<exists>x\<in>s - {a}. x j \<noteq> 0"   and "\<forall>j<n. \<exists>x\<in>s - {a}. x j \<noteq> p" shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 2"
dom12_d: fixes x :: "'a" shows "d x = 1\<^sub>\<sigma> \<sqinter> x \<cdot> 1\<^sub>\<pi> \<parallel> nc"
sup_commute: fixes x :: "'g multiset"   and y :: "'g multiset" shows "x \<union># y = y \<union># x"
ConsNil: fixes e' :: "'a \<times> 'a" shows "P e w [e']"
elts_vcard_vsum_eqpoll: fixes i :: "V"   and j :: "V" shows "elts (vcard (i \<Uplus> j)) \<approx> ZFC_Cardinals.Inl ` elts i \<union> ZFC_Cardinals.Inr ` elts j"
seqE: fixes g :: "J.Arr"   and f :: "J.Arr" assumes "J.seq g f" obtains "J.arr f" and "J.arr g" and "J.dom g = J.cod f"
mpoly_map_vars_Const: fixes f :: "nat \<Rightarrow> nat"   and c :: "'a" shows "mpoly_map_vars f (Const c) = Const c"
stk_strict_subs_def: fixes S :: "'c set" shows "stk_strict_subs S = (S \<noteq> UNIV)"