hcju/isabelleps huggingface dataset:
worklist_tree_aux_rule2: fixes succs :: "'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and ws :: "'c list"   and s :: "'d"   and ws' :: "'c list"   and s' :: "'d"   and I :: "'c \<Rightarrow> bool"   and S :: "'d \<Rightarrow> bool"   and P :: "'c list \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "worklist_tree_aux succs f (ws, s) = Some (ws', s')"   and "invariant I succs"   and "\<forall>x\<in>set ws. I x"   and "S s"   and "\<And>x s. \<lbrakk>I x; S s\<rbrakk> \<Longrightarrow> S (f x s)"   and "\<And>s. P [] s s"   and "\<And>r x ws s. \<lbrakk>I x; \<forall>x\<in>set ws. I x; S s; P ws (f x s) r\<rbrakk> \<Longrightarrow> P (x # ws) s r" shows "\<exists>rs. set rs = (Rel succs)\<^sup>* `` set ws \<and> P rs s s'"
m1: shows "m1 = \<lparr>iiface = iif1, oiface = oif1, src = sip1, dst = dip1, proto = p1, sports = sps1, dports = dps1\<rparr>"
Sup_eq_top_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Sup A = top) = (\<forall>x. less x top \<longrightarrow> (\<exists>i\<in>A. less x i))"
subset_Image1_Image1_iff: fixes r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "Preorder r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(r `` {a} \<subseteq> r `` {b}) = ((b, a) \<in> r)"
proj_lnull: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "cnf llist" shows "(dynamic_component.proj tCMP active c t = []\<^sub>l) = (\<forall>k\<in>lset t. \<not> active c k)"
set_tag_name_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_tag_name_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h = type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h'"
list_all2_reorder_left_invariance: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and xs' :: "'c list" assumes "list_all2 R xs ys"   and "mset xs' = mset xs" shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
\<sigma>s_\<mu>'_2: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j \<le> i" shows "rat_of_int (\<mu>' i j) = gs.\<mu>' i j"
nn_integral_measure_spmf: fixes p :: "'a spmf"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_spmf p) f = (\<Sum>\<^sup>+ x. ennreal (spmf p x) * f x)"
module_axioms_def: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme" shows "module_axioms R M \<equiv> ((\<forall>a x. a \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M) \<and> (\<forall>a b x. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> (a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x)) \<and> (\<forall>a x y. a \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> y \<in> carrier M \<longrightarrow> a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y) \<and> (\<forall>a b x. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)) \<and> (\<forall>x. x \<in> carrier M \<longrightarrow> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = x)"
lincomb_def: fixes a :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "N.lincomb a A = (\<Oplus>\<^bsub>md N\<^esub>v\<in>A. a v \<odot>\<^bsub>md N\<^esub> v)"
in_hhom_def: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>S b\<guillemotright> \<equiv> S.arr \<mu> \<and> S.src \<mu> = a \<and> S.trg \<mu> = b"
enum_alt_bool: shows "enum_alt \<equiv> alt_from_ord [False, True]"
pdg_tiny_digraph_dg_prod: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "tiny_digraph \<beta> (dg_prod I \<AA>)"
delete_eq: fixes k :: "'a" shows "delete k = filter (\<lambda>(k', uu_). k \<noteq> k')"
intros_1: fixes a :: "'c set"   and A :: "'c set set"   and sp :: "'c set" assumes "a \<in> A" shows "a \<in> sigma_sets sp A"
comp_bounded_clinear: fixes A :: "'a \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'a" assumes "bounded_clinear A"   and "bounded_clinear B" shows "bounded_clinear (A \<circ> B)"
write_cpu_tt_def: fixes w :: "word8" shows "write_cpu_tt w \<equiv> do tbr_val \<leftarrow> gets (cpu_reg_val TBR); new_tbr_val \<leftarrow> gets (\<lambda>s. write_tt w tbr_val); write_cpu new_tbr_val TBR; return () od"
filters_def: shows "filters = {F. F \<noteq> {} \<and> (\<forall>a b. a \<in> F \<and> b \<in> F \<longrightarrow> a * b \<in> F) \<and> (\<forall>a b. a \<in> F \<and> a \<le> b \<longrightarrow> b \<in> F)}"
simps_2: fixes v :: "'a"   and va :: "'a list"   and p2 :: "'a list" shows "R.poly_mult (v # va) p2 = R.poly_add (map ((\<otimes>) (lead_coeff (v # va))) p2 @ replicate (degree (v # va)) \<zero>) (R.poly_mult (tl (v # va)) p2)"
times_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A ** B = {a. \<exists>x\<in>A. \<exists>y\<in>B. a = x * y}"
no_checkin_no_newkey: fixes s\<^sub>2 :: "event list"   and g :: "guest"   and r :: "room"   and k :: "key"   and k' :: "key"   and s\<^sub>1 :: "event list"   and k'' :: "key"   and g' :: "guest" assumes "hotel (s\<^sub>2 @ [Check_in g r (k, k')] @ s\<^sub>1)"   and "no_Check_in s\<^sub>2 r" shows "(k', k'') \<notin> cards (s\<^sub>2 @ Check_in g r (k, k') # s\<^sub>1) g'"
Suc: shows "stake n ` streams UNIV = gener_bool_list n"
eq: shows "matrix_sum d (\<lambda>l. tensor_P (proj_psi_l l) (proj_k l)) (R + 1) = tensor_P (proj_psi_l 0) (proj_k 0) + matrix_sum d (\<lambda>l. tensor_P (proj_psi_l (l + 1)) (proj_k (l + 1))) R"
mk_total_program_def: fixes args :: "'b set \<times> ('b \<times> 'b) set set \<times> ('b \<times> 'b) set set" shows "mk_total_program args \<equiv> totalize (mk_program args)"
error_abs_def: shows "error_abs \<equiv> PRJ('b\<cdot>'a error) oo EMB('a\<^sub>\<bottom> \<oplus> 'b\<^sub>\<bottom>)"
s2p_inter_hom: fixes A :: "'b set"   and B :: "'b set" shows "s2p (A \<inter> B) = s2p A \<sqinter> s2p B"
embed_measure_count_space': fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "embed_measure (count_space A) f = count_space (f ` A)"
assms_2: shows "sub_set_system \<V> \<B> \<U> \<A>"
composite_functor_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
fls_integral_mult_const_right: fixes f :: "'a fls"   and c :: "'a" shows "fls_integral (f * fls_const c) = fls_integral f * fls_const c"
lconjby_rconjby: fixes x :: "'a"   and y :: "'a" shows "lconjby x (rconjby x y) = y"
weakCase: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and \<phi> :: "'c"   and CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "(\<phi>, P) mem CsP"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P"   and "insertAssertion (extractFrame R) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"   and "insertAssertion (extractFrame R) \<Psi> \<hookrightarrow>\<^sub>F \<langle>\<epsilon>, \<Psi>\<rangle>" shows "\<Psi> : R \<rhd> Cases CsP \<Longrightarrow>\<alpha> \<prec> P'"
chamber_in_subcomplex: fixes Y :: "'a set set"   and C :: "'a set" assumes "folding.ChamberSubcomplex Y"   and "C \<in> Y"   and "folding.chamber C" shows "SimplicialComplex.maxsimp Y C"
back_subst: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "P a"   and "a = b" shows "P b"
l_adopt_node_get_shadow_root_axioms: shows "l_adopt_node_get_shadow_root adopt_node_locs get_shadow_root_locs"
rW: shows "dim_row (mat_of_rows n (map vec_inv vs)) = l"
elementary_monoidal_category_axioms: shows "elementary_monoidal_category (\<cdot>\<^sub>S) (\<otimes>\<^sub>S) \<I> (\<lambda>a. a) (\<lambda>a. a) assoc\<^sub>S"
IS: fixes xs :: "'a list"   and k :: "nat"   and x :: "'a" assumes "P xs k" shows "P (x # xs) (Suc k)"
netlimit_at_within_mono: fixes t :: "'b"   and T :: "'b set"   and S :: "'b set" assumes "t \<in> interior T"   and "T \<subseteq> S" shows "netlimit (at t within S) = t"
except_keys_Int: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and U :: "'a set" shows "except p (keys p \<inter> U) = except p U"
approximate_sets_ode_def: fixes set_of_appr :: "'a list \<Rightarrow> real list set"   and appr_rell :: "('a list \<times> real list set) set"   and optns :: "'a numeric_options"   and ops :: "'a approximate_set_ops" shows "approximate_sets_ode set_of_appr appr_rell optns ops \<equiv> approximate_sets ops set_of_appr appr_rell optns"
valid_cfg_action_in_K: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "action cfg \<in> K (state cfg)"
strategy_winning_by_Odd_def: fixes \<sigma>\<^sub>o :: "'a list \<Rightarrow> 'a"   and p :: "'a list" shows "strategy_winning_by_Odd \<sigma>\<^sub>o p \<equiv> \<forall>\<sigma>\<^sub>e. induced_play (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) p \<notin> A"
bind_mono_strong: fixes m :: "('a, 'b) state"   and f :: "'b \<Rightarrow> ('a, 'c) state" assumes "mono_state m"   and "\<And>x s s'. run_state m s = (x, s') \<Longrightarrow> mono_state (f x)" shows "mono_state (m \<bind> f)"
Diff_Int_distrib2: fixes A :: "'h set"   and B :: "'h set"   and C :: "'h set" shows "(A - B) \<inter> C = A \<inter> C - B \<inter> C"
twoCisTypeI_3: shows "cubeImage twoC = {(x, y). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
isObjV_simps_5: fixes c :: "CTypeId"   and a :: "nat" shows "isObjV (objV c a) = True"
elements_mat_four_block_mat_supseteq: fixes A1 :: "'a mat"   and n :: "nat"   and A2 :: "'a mat"   and m :: "nat"   and A3 :: "'a mat"   and A4 :: "'a mat" assumes "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat n m"   and "A3 \<in> carrier_mat m n"   and "A4 \<in> carrier_mat m m" shows "elements_mat A1 \<union> elements_mat A2 \<union> elements_mat A3 \<union> elements_mat A4 \<subseteq> elements_mat (four_block_mat A1 A2 A3 A4)"
somegcd_meet: fixes a :: "'c"   and G :: "('c, 'd) monoid_scheme"   and b :: "'c" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "somegcd G a b = a \<sqinter>\<^bsub>division_rel G\<^esub> b"
N: fixes n :: "nat" assumes "N \<le> n" shows "real n * norm x ^ n < r ^ n"
P1: shows "Bet A' B' C'' \<and> Cong B' C'' B C"
cbox_borel: fixes a :: "'b"   and b :: "'b" shows "cbox a b \<in> sets borel"
comp_apply: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(f \<circ> g) x = f (g x)"
interchange_same_rows: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and a :: "'c" shows "interchange_rows A a a = A"
put_writer_def: fixes put :: "'l \<Rightarrow> 'k \<Rightarrow> 'k"   and s :: "'l"   and m :: "('i, 'j, 'k) writerT" shows "put_writer put s m = WriterT (put s (run_writer m))"
asubg_subset: fixes H :: "'a set" assumes "A +> H" shows "H \<subseteq> carrier A"
decseq_imp_monoseq: fixes X :: "nat \<Rightarrow> 'a" assumes "decseq X" shows "monoseq X"
lb2: shows "\<forall>x\<le>lb2. sgn (poly p x) = sgn_neg_inf p"
simps_17: shows "op_sem Fst = fst \<circ> extract_pair"
chi_def: shows "chi = (\<lambda>val. AND {atom\<^sub>p(p) |p. p \<in> val} and\<^sub>p AND {not\<^sub>p atom\<^sub>p(p) |p. p \<in> A - val})"
check_poly_weak_anti_mono_discrete_def: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" shows "check_poly_weak_anti_mono_discrete p v \<equiv> check_poly_ge p (poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'d), PVar v] else PVar w)) p)"
N_def: shows "N \<equiv> distr M (stock_measure INTEG) (lift_RealIntVal2 f f')"
Ostrowski_base_def: fixes K :: "('c, 'd) Ring_scheme"   and vv :: "nat \<Rightarrow> 'c \<Rightarrow> ant"   and n :: "nat" shows "\<Omega>\<^bsub>K vv n\<^esub> = (\<lambda>j\<in>{h. h \<le> n}. SOME x. x \<in> carrier K \<and> Ostrowski_elem K n (vv \<circ> \<tau>\<^bsub>0 j\<^esub>) x)"
assms_2: shows "y = QCOERCE['d\<^sub>2] x"
HSND: fixes a :: "tm"   and b :: "tm"   and c :: "tm"   and d :: "tm" shows "{HPair a b EQ HPair c d} \<turnstile> b EQ d"
f0: fixes x :: "'b" assumes "x \<in> b" shows "f x = (0::'c)"
merge_guards_simps_7: fixes p :: "'e" shows "merge_guards (Call p) = Call p"
wW: shows "w = #(#w + Polynomial.smult (of_nat q) (#w'))"
simps_1: fixes i :: "int" shows "\<gamma>_const (Const i) = {i}"
eqvtChainEq: fixes P :: "pi"   and P' :: "pi"   and perm :: "(name \<times> name) list" shows "P \<Longrightarrow>\<^sub>\<tau> P' = perm \<bullet> P \<Longrightarrow>\<^sub>\<tau> perm \<bullet> P'"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "split_min_tm \<langle>l, x, r\<rangle> =1 (if l = \<langle>\<rangle> then return (x, r) else split_min_tm l \<bind> (\<lambda>(y, l'). return (y, \<langle>l', x, r\<rangle>)))"
SUP_mono': fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` A))"
graph_plan_lemma_17: fixes as_1 :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and as_2 :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and s :: "('c, 'd) fmap" assumes "as_1 @ as_2 = as"   and "sat_precond_as s as" shows "sat_precond_as s as_1 \<and> sat_precond_as (exec_plan s as_1) as_2"
map_mat_transpose: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b mat" shows "(map_mat f A)\<^sup>T = map_mat f A\<^sup>T"
elem: fixes x :: "'a"   and xs :: "'a list" shows "ListMem x (x # xs)"
x'_def: shows "x' = fst (normalize_quot (x, y))"
lift_terms_NF: fixes ts :: "dB list"   and i :: "nat" assumes "listall NF ts"   and "listall (\<lambda>t. \<forall>i. NF (lift t i)) ts" shows "listall NF (map (\<lambda>t. lift t i) ts)"
coplanar_perm_7: 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 B A D C"
finite_cone_iff_cone_list: fixes Vs :: "'a vec set"   and Vsl :: "'a vec list" assumes "Vs \<subseteq> carrier_vec n"   and "Vs = set Vsl" shows "n.finite_cone Vs = n.cone_list Vsl"
subset_not_subset_eq: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subset> B) = (A \<subseteq> B \<and> \<not> B \<subseteq> A)"
assms_1: shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
eq_defined: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a"   and y :: "'a" assumes "class.Eq_def eq"   and "x \<noteq> \<bottom>"   and "y \<noteq> \<bottom>" shows "eq\<cdot>x\<cdot>y \<noteq> \<bottom>"
simps_3: fixes x21 :: "'c"   and x22 :: "'c list" shows "x21 # x22 \<noteq> []"
from_hma\<^sub>m_def: fixes a :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>m a = Matrix.mat CARD('c) CARD('b) (\<lambda>(i, j). a $h Bij_Nat.from_nat i $h Bij_Nat.from_nat j)"
euler_liar_is_fermat_liar: assumes "1 < n"   and "euler_liar (int a) n"   and "coprime a n"   and "odd n" shows "local.fermat_liar"
m_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H" shows "x \<otimes>\<^bsub>H\<^esub> y \<in> carrier H"
DVDJ_DVD: fixes x :: "real"   and n :: "int"   and bs :: "real list"   and i :: "int"   and s :: "MIR.num" assumes "0 \<le> x"   and "x < 1"   and "0 < real_of_int n" shows "Ifm (x # bs) (DVDJ i n s) = Ifm (x # bs) (Dvd i (CN 0 n s))"
AbstrLevels_S6_A31: fixes i :: "AbstrLevelsID" assumes "sS6 \<in> AbstrLevel i" shows "sA31 \<notin> AbstrLevel i"
odd_of_bool_self: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "bool" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (zero_neq_one.of_bool one zero p)) = p"
DS': fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "DS \<rho> \<in> carrier_mat d d \<and> partial_density_operator (DS \<rho>)"
ltl_lunstream: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "ltl (lunstream g s) = (case g s of Done \<Rightarrow> LNil | Skip s' \<Rightarrow> ltl (lunstream g s') | Yield x s' \<Rightarrow> lunstream g s')"
construct_NofI_prop_R: fixes p :: "real poly"   and I1 :: "real poly list"   and I2 :: "real poly list" assumes "p \<noteq> 0" shows "construct_NofI_R p I1 I2 = rat_of_int (int (card {x. poly p x = 0 \<and> (\<forall>q\<in>set I1. poly q x = 0) \<and> 0 < poly (prod_list I2) x}) - int (card {x. poly p x = 0 \<and> (\<forall>q\<in>set I1. poly q x = 0) \<and> poly (prod_list I2) x < 0}))"
neighbors_in_V: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and v :: "nat" assumes "discharge_invar c s t u fo lo no f l n"   and "v \<in> n u" shows "v \<in> Graph.V c"
tinc_correct: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and n :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and lm' :: "nat list" assumes "ly = layout_of ap"   and "inv_locate_a (as, lm) (n, l, r) ires"   and "lm' = abc_lm_s lm n (Suc (abc_lm_v lm n))" shows "\<exists>stp l' r'. steps0 (Suc 0, l, r) tinc_b stp = (10, l', r') \<and> inv_stop (as, lm') (10, l', r') ires"
predicate2D: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "P \<le> Q"   and "P x y" shows "Q x y"
correctly_synchronizedD: fixes P :: "'a prog"   and \<E> :: "('b \<times> ('c, 'b) obs_event action) llist set"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "correctly_synchronized P \<E>"   and "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "sequentially_consistent P (E, ws)" shows "\<forall>a a'. a \<in> actions E \<longrightarrow> a' \<in> actions E \<longrightarrow> P,E \<turnstile>a\<dagger>a' \<longrightarrow> P,E \<turnstile> a \<le>hb a' \<or> P,E \<turnstile> a' \<le>hb a"
singleton: fixes x :: "'a" shows "semilattice_sup.Sup_fin (\<squnion>1) {x} = x"
subsuming_states_map_add: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and s :: "'a \<Rightarrow> 'b option" assumes "\<And>x. x \<in> dom m1 \<inter> dom m2 \<Longrightarrow> m1 x = m2 x" shows "(m1 ++ m2 \<subseteq>\<^sub>m s) = (m1 \<subseteq>\<^sub>m s \<and> m2 \<subseteq>\<^sub>m s)"
split_float_interval_def: fixes x :: "float interval" shows "split_float_interval x = split_interval x ((lower x + upper x) * Float 1 (- 1))"
residual_eq_iff_perm_renaming: fixes \<alpha>1 :: "'a"   and P1 :: "'b"   and \<alpha>2 :: "'a"   and P2 :: "'b" shows "(\<langle>\<alpha>1,P1\<rangle> = \<langle>\<alpha>2,P2\<rangle>) = (\<exists>p. supp (\<alpha>1, P1) - bn \<alpha>1 = supp (\<alpha>2, P2) - bn \<alpha>2 \<and> (supp (\<alpha>1, P1) - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> (\<alpha>1, P1) = (\<alpha>2, P2) \<and> p \<bullet> bn \<alpha>1 = bn \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1)"
mult_1_left: fixes a :: "'a" shows "(0::'a) + a = a"
seq_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.seq g f = (B.VVV.arr f \<and> B.VVV.arr g \<and> B.VxVxV.seq g f)"
run_bind: fixes m :: "'c + 'd"   and f :: "'d \<Rightarrow> 'c + 'e" assumes "isOK m" shows "projr (m \<bind> f) = projr (f (projr m))"
enat_cocase_eSuc: fixes z :: "'a"   and s :: "enat \<Rightarrow> 'a"   and n :: "enat" shows "(case eSuc n of 0 \<Rightarrow> z | eSuc x \<Rightarrow> s x) = s n"
wfp: fixes AP' :: "('U, 'C, 'Out) Noninterference.aprop set"   and \<pi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream" shows "wfp AP' \<pi> = (\<forall>i. fst (\<pi> !! i) \<in> UNIV \<and> snd (\<pi> !! i) \<subseteq> AP' \<and> snd (\<pi> !! i) \<inter> UNIV = L (fst (\<pi> !! i)) \<and> fst (\<pi> !! Suc i) \<in> \<delta> (fst (\<pi> !! i)))"
measure_pmf_posI: fixes x :: "'h"   and p :: "'h pmf"   and A :: "'h set" assumes "x \<in> set_pmf p"   and "x \<in> A" shows "0 < measure_pmf.prob p A"
rep_eq: fixes x :: "('b, 'a, 'c) Vardi.var" shows "set2_var x = set2_var0 (rep_var x)"
LeftDerivationLadder_exists: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and \<gamma> :: "symbol list"   and j :: "nat" assumes "LeftDerivation a D \<gamma>"   and "is_sentence \<gamma>"   and "j < length \<gamma>" shows "\<exists>L. LeftDerivationLadder a D L \<gamma> \<and> ladder_last_j L = j"
assms_2: shows "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt3"
assms: shows "c_sources I D u ys \<subseteq> c_sources I D u zs"
simps_1: fixes m :: "int"   and i :: "int"   and k :: "int"   and ks :: "int list" shows "hd_coeff1 m (Le i (k # ks)) = (if k = 0 then Le i (k # ks) else let m' = m div \<bar>k\<bar> in Le (m' * i) (sgn k # m' *\<^sub>s ks))"
ins_list_sorted: fixes xs :: "'b list"   and a :: "'b"   and x :: "'b"   and ys :: "'b list" assumes "sorted (xs @ [a])" shows "ins_list x (xs @ a # ys) = (if x < a then ins_list x xs @ a # ys else xs @ ins_list x (a # ys))"
game4_def: shows "game4 = (\<lambda>\<A>. key_gen \<bind> (\<lambda>key. exec_gpv (oracle_encrypt4 key \<oplus>\<^sub>O oracle_decrypt2 key) \<A> () \<bind> (\<lambda>(b', uu_). map_spmf ((=) b') coin_spmf)))"
cl_in_rank_of: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> \<E>"   and "X \<subseteq> \<E>"   and "x \<in> \<E>.cl_in \<E> X" shows "\<E>.rank_of (insert x X) = \<E>.rank_of X"
ResInput: fixes x :: "name"   and a :: "name"   and y :: "name"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> y" shows "<\<nu>x>a<y>.P \<equiv>\<^sub>e a<y>.(<\<nu>x>P)"
vseq_implies_hpar_1: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.seq \<nu> \<mu>" shows "S.src \<nu> = S.src \<mu>"