hcju/isabelleps huggingface dataset:
if_loc_reach: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c' :: "('b, 'a, 'c) Stmt"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'd"   and e :: "'c"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and annos :: "'b ModeUpd list"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'd" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "((c', mds'), mem') \<in> sifum_security.loc_reach (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((sifum_modes.annotate (Stmt.If e c\<^sub>1 c\<^sub>2) annos, mds), mem)" shows "c' = sifum_modes.annotate (Stmt.If e c\<^sub>1 c\<^sub>2) annos \<and> mds' = mds \<or> (\<exists>mem''. ((c', mds'), mem') \<in> sifum_security.loc_reach (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((c\<^sub>1, sifum_modes.update_annos mds annos), mem'')) \<or> (\<exists>mem''. ((c', mds'), mem') \<in> sifum_security.loc_reach (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((c\<^sub>2, sifum_modes.update_annos mds annos), mem''))"
strongBisimWeakBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sub>e Q" shows "P \<approx> Q"
simps_29: fixes f1 :: "String.literal \<Rightarrow> typ \<Rightarrow> 'a"   and f2 :: "variable \<Rightarrow> typ \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'a"   and f4 :: "typ \<Rightarrow> Core.term \<Rightarrow> 'a"   and f5 :: "Core.term \<Rightarrow> Core.term \<Rightarrow> 'a"   and x41 :: "typ"   and x42 :: "Core.term" shows "(case Abs x41 x42 of Ct x xa \<Rightarrow> f1 x xa | Fv x xa \<Rightarrow> f2 x xa | Bv x \<Rightarrow> f3 x | Abs x xa \<Rightarrow> f4 x xa | x $ xa \<Rightarrow> f5 x xa) = f4 x41 x42"
ideal_subset: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set"   and h :: "'d" assumes "Ring R"   and "ideal R I"   and "h \<in> I" shows "h \<in> carrier R"
lagrange_unique: fixes zs_ws :: "(complex \<times> complex) list"   and p1 :: "complex poly"   and p2 :: "complex poly" assumes "0 < length zs_ws"   and "distinct (map fst zs_ws)"   and "degree p1 \<le> length zs_ws - 1 \<and> (\<forall>x y. (x, y) \<in> set zs_ws \<longrightarrow> poly p1 x = y)"   and "degree p2 \<le> length zs_ws - 1 \<and> (\<forall>x y. (x, y) \<in> set zs_ws \<longrightarrow> poly p2 x = y)" shows "p1 = p2"
prefix_to_wordinterval_set_eq: fixes pfx :: "'b prefix_match" shows "wordinterval_to_set (prefix_to_wordinterval pfx) = prefix_to_wordset pfx"
apex_composite: shows "\<nu>\<pi>.apex = \<nu>.leg0 \<down>\<down> \<pi>.leg1"
assms: shows "\<not> b =simp=> B \<subseteq> A"
posr: fixes x :: "'c"   and y :: "'c" assumes "D x y"   and "x \<oplus> y \<in> E" shows "y \<in> E"
freshEnv_updEnv_I: fixes ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "ys \<noteq> xs \<or> y \<noteq> x"   and "fresh ys y X"   and "rho ys y = None"   and "\<And>zs z Z. \<lbrakk>zs \<noteq> xs \<or> z \<noteq> x; rho zs z = Some Z\<rbrakk> \<Longrightarrow> fresh ys y Z" shows "freshEnv ys y (rho [x \<leftarrow> X]_xs)"
"2_3": fixes x :: "real \<times> real" assumes "x \<in> set (y # ys)" shows "Counterclockwise_2D_Arbitrary.lex x 0"
unstream_fix_gen: fixes g :: "'f \<Rightarrow> ('e, 'g) generator"   and a :: "'f"   and s :: "'g" shows "unstream (fix_gen g) (a, s) = unstream (g a) s"
not_interp_to_Interp_imp_le: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and A :: "'b"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and D :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "A \<notin> ground_resolution_with_selection.interp S N C"   and "A \<in> ground_resolution_with_selection.Interp S N D" shows "C \<le> D"
equivalence_def: fixes S :: "('c, 'd) eq_object_scheme" shows "equivalence S \<equiv> (\<forall>x. x \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y. x .=\<^bsub>S\<^esub> y \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> y .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y z. x .=\<^bsub>S\<^esub> y \<longrightarrow> y .=\<^bsub>S\<^esub> z \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> z \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> z)"
simps_8: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "Sequence \<tau> \<squnion> \<sigma> = (case \<sigma> of Collection \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Set \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | OrderedSet \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Bag \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Sequence \<rho> \<Rightarrow> Sequence (\<tau> \<squnion> \<rho>) | _ \<Rightarrow> OclSuper)"
Z: shows "Complete_Partial_Order.chain (fun_ord (\<le>)) Z"
of_int_mult: fixes 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 w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w * z) = times (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
lossless_init: fixes h :: "'a"   and w :: "nat" shows "lossless_spmf (init h w)"
of_nat_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 1 = one"
sum_pos: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> {#} \<subset># f i" shows "{#} \<subset># comm_monoid_add.sum (+) {#} f I"
trace_eq_sim: shows "(local.valid_insecQ <+> nlists UNIV (id' \<eta>) <+> UNIV) \<turnstile>\<^sub>R RES (callee_auth_channel sim \<oplus>\<^sub>O \<dagger>\<dagger>channel.send_oracle \<oplus>\<^sub>O \<dagger>\<dagger>channel.recv_oracle) (Inl None, (), Void) \<approx> RES (local.lazy_channel_insec \<oplus>\<^sub>O lazy_channel_send \<oplus>\<^sub>O local.lazy_channel_recv_f) (Void, None, Map.empty)"
stabiliser_subgroup_corollary: fixes g :: "Vertex \<Rightarrow> Vertex"   and h :: "Vertex \<Rightarrow> Vertex"   and x :: "Vertex" assumes "g \<in> carrier tetrahedral_group"   and "h \<in> carrier tetrahedral_group" shows "(apply_rotation g x = apply_rotation h x) = (inv\<^bsub>tetrahedral_group\<^esub> g \<otimes>\<^bsub>tetrahedral_group\<^esub> h \<in> tetrahedral.stabiliser x)"
agreeR: shows "Vagree \<nu> \<nu>' (\<Union> (primify ` FVT t2))"
Un_Diff_cancel: fixes A :: "'e set"   and B :: "'e set" shows "A \<union> (B - A) = A \<union> B"
numeral_times_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "num"   and n :: "num" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m * n)"
ALG_P_erwischt_alle: fixes init :: "'c list"   and sws :: "nat list" assumes "distinct init"   and "\<forall>l<length sws. Suc (sws ! l) < length init" shows "length sws = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALG_P sws x y init)"
prime_int_iff: fixes n :: "int" shows "prime n = (1 < n \<and> (\<forall>m. 0 \<le> m \<and> m dvd n \<longrightarrow> m = 1 \<or> m = n))"
length_0_conv: fixes xs :: "'b list" shows "(\<^bold>|xs\<^bold>| = 0) = (xs = \<epsilon>)"
scalar_prod_uminus_left: fixes v :: "'a vec"   and w :: "'a vec" assumes "dim_vec v = dim_vec w" shows "- v \<bullet> w = - (v \<bullet> w)"
msetext_huet_wf: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. msetext_huet gt ys xs)"
ordering3: shows "\<forall>n n' n''. (finite {a, b} \<longrightarrow> n'' < card {a, b}) \<and> n < n' \<and> n' < n'' \<longrightarrow> ord (if n = 0 then a else b) (if n' = 0 then a else b) (if n'' = 0 then a else b)"
spr_repSim_simps_2: fixes t :: "('a, 'es, 'as) BEState Trace" shows "spr_repSim (spr_sim t) = (tFirst t, tLast t)"
rel_fun_conversep: fixes A :: "'j \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'k \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun A\<inverse>\<inverse> B\<inverse>\<inverse> = (rel_fun A B)\<inverse>\<inverse>"
intro: fixes get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" assumes "l_get_host get_host get_host_locs"   and "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes get_root_node get_owner_document" shows "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_owner_document"
exhaust: fixes y :: "('b, 'c) RBT_Impl.rbt" obtains "y = rbt.Empty"   | x21 :: "color" and x22 :: "('b, 'c) RBT_Impl.rbt" and x23 :: "'b" and x24 :: "'c" and x25 :: "('b, 'c) RBT_Impl.rbt" where "y = Branch x21 x22 x23 x24 x25"
ext: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.comp g f \<noteq> VV.null" shows "VV.seq g f"
assms_13: fixes a :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and b :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "a \<in> set hs"   and "b \<in> set gs \<union> set bs \<union> set hs"   and "(a, b) \<in>\<^sub>p set (ap gs bs ps hs data)"   and "fst a \<noteq> 0"   and "fst b \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)"
unifiert_unifier\<^sub>l\<^sub>s: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" assumes "unifier\<^sub>t\<^sub>s \<sigma> (literal_to_term ` L)" shows "unifier\<^sub>l\<^sub>s \<sigma> L"
finite_enumerate_in_set: fixes S :: "'a set"   and n :: "nat" assumes "finite S"   and "n < card S" shows "wellorder_class.enumerate S n \<in> S"
is_regular_spair_spp_def: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "is_regular_spair_spp p q = (snd p \<noteq> 0 \<and> snd q \<noteq> 0 \<and> punit.lt (snd q) \<oplus> fst p \<noteq> punit.lt (snd p) \<oplus> fst q)"
igAbsIPresIGWlsSTR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igAbsIPresIGWlsSTR MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X) = igWls MOD s X"
inv_cancelL_2: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "cring R"   and "x \<in> Units R"   and "y \<in> carrier R"   and "z \<in> carrier R"   and "y = z \<otimes>\<^bsub>R\<^esub> x" shows "y \<otimes>\<^bsub>R\<^esub> inv\<^bsub>R\<^esub> x = z"
s: shows "s = Fun f (ss1 @ D\<langle>t\<rangle> # ss2)"
elims_2: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b list" assumes "successive x xa" obtains uu_ :: "'b \<Rightarrow> bool" where "x = uu_" and "xa = []"   | P :: "'b \<Rightarrow> bool" and uv_ :: "'b" where "x = Pa__" and "xa = [uv_]"   | P :: "'b \<Rightarrow> bool" and xb :: "'b" and y :: "'b" and xs :: "'b list" where "x = P" and "xa = xb # y # xs" and "\<not> P y \<and> successive P xs \<or> \<not> P xb \<and> successive P (y # xs)"
thread_oks_bisim_inv: fixes ts1 :: "'t \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and ts2 :: "'t \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and tas1 :: "('t, 'x1, 'm1) new_thread_action list"   and tas2 :: "('t, 'x2, 'm2) new_thread_action list" assumes "\<forall>t. (ts1 t = None) = (ts2 t = None)"   and "list_all2 (nta_bisim bisim) tas1 tas2" shows "thread_oks ts1 tas1 = thread_oks ts2 tas2"
Sup_pres_iso: fixes f :: "'b \<Rightarrow> 'c" assumes "Sup_pres f" shows "mono f"
assms_1: shows "group_homomorphism \<eta> A multA oneA B multB oneB"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_ring_hom hom" shows "idom_hom hom"
nxtActLe: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. active c (t i)" shows "n \<le> dynamic_component.nxtAct active c t n"
fimsub: shows "f ` ball a r \<subseteq> - f ` sphere a r"
less_trans: fixes x :: "'b Word.word"   and y :: "'b Word.word"   and z :: "'b Word.word" assumes "x <s y"   and "y <s z" shows "x <s z"
in_span_delete: fixes a :: "'b"   and S :: "'b set"   and b :: "'b" assumes "a \<in> span S"   and "a \<notin> span (S - {b})" shows "b \<in> span (insert a (S - {b}))"
set_notEmptyE: fixes S :: "'a set" assumes "S \<noteq> {}" obtains x :: "'a" where "x \<in> S"
gen_model_rels_worlds: fixes w' :: "'e"   and w'' :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and w :: "'e"   and a :: "'f" assumes "(w', w'') \<in> relations (gen_model M w) a" shows "w' \<in> worlds (gen_model M w) \<and> w'' \<in> worlds (gen_model M w)"
order_iff: fixes a :: "'r"   and b :: "'r" shows "(a \<le> b) = (a = min a b)"
nth_prime_le_prime_sequence: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "strict_mono_on p {..<n}"   and "\<And>k. k < n \<Longrightarrow> prime (p k)"   and "k < n" shows "nth_prime k \<le> p k"
abs_ineq_iff: fixes H :: "'c set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and h :: "'c \<Rightarrow> real" assumes "Subspace.subspace H E"   and "vectorspace E"   and "seminorm E p"   and "linearform H h" shows "(\<forall>x\<in>H. \<bar>h x\<bar> \<le> p x) = (\<forall>x\<in>H. h x \<le> p x)"
set_mult_sub_sum_mult: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier R"   and "ideal R B" shows "set_mult R A B \<subseteq> sum_mult R A B"
FiniteSends: fixes p :: "'p"   and s :: "'s"   and m :: "'v messageValue" shows "finite {v. v \<in># sends p s m}"
set_lebesgue_integral_cong: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "A \<in> sets M"   and "\<forall>x. x \<in> A \<longrightarrow> f x = g x" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A g"
cf_pos_poly_main: fixes f :: "int poly" shows "Polynomial.smult (sgn (lead_coeff f) * content f) (cf_pos_poly f) = f"
trace_at_simp: fixes s :: "'d"   and t' :: "real"   and s' :: "'d"   and \<omega> :: "(real \<times> 'd) stream"   and j :: "real" shows "trace_at s ((t', s') ## \<omega>) j = (if t' \<le> j then trace_at s' \<omega> j else s)"
xor_left_commute: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl b (boolean_algebra.xor conj disj compl a c) = boolean_algebra.xor conj disj compl a (boolean_algebra.xor conj disj compl b c)"
mrDenyAll_is_unique: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and r :: "((int \<times> int) set set, int) Combinators" assumes "wellformed_policy1_strong p"   and "applied_rule_rev C x p = \<lfloor>DenyAll\<rfloor>"   and "r \<in> set (tl p)" shows "x \<notin> dom (C r)"
de_morgans_1: fixes a :: "trilean"   and b :: "trilean" shows "\<not>? a \<or>? b = (\<not>? a) \<and>? (\<not>? b)"
bin_nth_drop_bit_iff: fixes n :: "nat"   and c :: "int"   and k :: "nat" shows "bit (drop_bit n c) k = bit c (n + k)"
OrdI: fixes i :: "V" assumes "Transset i"   and "\<And>x. x \<in>\<^sub>\<circ> i \<Longrightarrow> Transset x" shows "Ord i"
LLL_reconstruction: fixes p :: "int"   and pl :: "int"   and f :: "int poly"   and us :: "int poly list"   and fs :: "int poly list"   and F :: "int poly"   and l :: "nat" assumes "LLL_implementation.LLL_reconstruction p pl f us = fs"   and "degree f \<noteq> 0"   and "poly_mod.unique_factorization_m pl f (lead_coeff f, mset us)"   and "f dvd F"   and "\<And>ui. ui \<in> set us \<Longrightarrow> poly_mod.Mp pl ui = ui"   and "F \<noteq> 0"   and "comm_monoid_mult_class.coprime (lead_coeff F) p"   and "poly_mod.square_free_m p F"   and "1 < pl"   and "pl = p ^ l"   and "prime p"   and "2 ^ (5 * (degree F - 1) * (degree F - 1)) * \<parallel>F\<parallel>\<^sup>2 ^ (2 * (degree F - 1)) < pl\<^sup>2" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
relation_rules_7: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c"   and Q :: "'c \<times> 'd \<Rightarrow> bool"   and c :: "'d" assumes "P (a, b)"   and "Q (b, c)" shows "(P ;; Q) (a, c)"
Quotient3_fset: shows "Quotient3 (\<approx>) abs_fset rep_fset"
strict_trans2: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a < b"   and "b \<le> c" shows "a < c"
bl_shiftr: fixes n :: "nat"   and w :: "'b word" assumes "n \<le> size w" shows "to_bl (w >> n) = replicate n False @ take (size w - n) (to_bl w)"
F: shows "offs_next ns ub (x # xs) index key mi ma 0 = ns ! i"
lift_trimmed_pos: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "trimmed_pos C" shows "trimmed_pos (MS (MSLCons (C, e) xs) f)"
simps: fixes x :: "'b"   and hs :: "'b hp list" shows "php (Hp x hs) = (\<forall>h\<in>set hs. Ball (set_hp h) ((\<le>) x) \<and> php h)"
bind_commute_pmf: fixes A :: "'e pmf"   and B :: "'f pmf"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> 'd pmf" shows "A \<bind> (\<lambda>x. B \<bind> C x) = B \<bind> (\<lambda>y. A \<bind> (\<lambda>x. C x y))"
division_product: fixes b :: "int"   and a :: "int" assumes "0 < b"   and "b \<le> a" shows "a div b = (a - b) div b + 1"
case_eq_if: fixes f1 :: "'x"   and f2 :: "'y \<Rightarrow> 'x"   and option :: "'y option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
TruePrefs_axioms: shows "TruePrefs Xd Xh Pd Ch x X ds Pd'"
poly_sgnx_left_right: fixes p :: "real poly"   and a :: "real" assumes "p \<noteq> 0" shows "sgnx (poly p) (at_left a) = (if even (order a p) then sgnx (poly p) (at_right a) else - sgnx (poly p) (at_right a))"
bound_intersect_2d_eq_None_iff: fixes prec :: "nat"   and Xs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real" shows "(bound_intersect_2d prec Xs x = None) = (\<forall>X\<in>set Xs. intersect_segment_xline prec X x = None)"
simps: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word" shows "pu w1 w2 = (if w1 = POS_INF then POS_INF else if w2 = POS_INF then POS_INF else if w1 = NEG_INF then if w2 = NEG_INF then NEG_INF else let sum = SCAST(32 \<rightarrow> 64) w2 + SCAST(32 \<rightarrow> 64) NEG_INF in if sum \<le>s SCAST(32 \<rightarrow> 64) NEG_INF then NEG_INF else SCAST(64 \<rightarrow> 32) sum else if w2 = NEG_INF then let sum = SCAST(32 \<rightarrow> 64) w1 + SCAST(32 \<rightarrow> 64) NEG_INF in if sum \<le>s SCAST(32 \<rightarrow> 64) NEG_INF then NEG_INF else SCAST(64 \<rightarrow> 32) sum else let sum = SCAST(32 \<rightarrow> 64) w1 + SCAST(32 \<rightarrow> 64) w2 in if SCAST(32 \<rightarrow> 64) POS_INF \<le>s sum then POS_INF else if sum \<le>s SCAST(32 \<rightarrow> 64) NEG_INF then NEG_INF else SCAST(64 \<rightarrow> 32) sum)"
nz: assumes "w1 = NEG_INF \<and> 0 = w2" shows "P w1 w2"
validVU: shows "isViewOn \<V> E\<^bsub>SES\<^esub>"
iT_Plus_neg_mem_iff: fixes x :: "nat"   and I :: "nat set"   and k :: "nat" shows "(x \<in> I \<oplus>- k) = (x + k \<in> I)"
list_decomp: fixes v :: "'a"   and p :: "'a list" assumes "v \<in> set p" shows "\<exists>xs ys. p = xs @ v # ys \<and> v \<notin> set xs"
prod_algebraE: fixes A :: "('d \<Rightarrow> 'e) set"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "A \<in> prod_algebra I M" obtains J :: "'d set" and E :: "'d \<Rightarrow> 'e set" where "A = prod_emb I M J (Pi\<^sub>E J E)" and "finite J" and "J \<noteq> {} \<or> I = {}" and "J \<subseteq> I" and "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
flow_undefined0: fixes t :: "real"   and x :: "'a" assumes "t \<notin> existence_ivl0 x" shows "flow0 x t = (0::'a)"
univalentD: fixes R :: "('d \<times> 'e) set"   and x :: "'d"   and y :: "'e"   and z :: "'e" assumes "univalent R"   and "(x, y) \<in> R"   and "(x, z) \<in> R" shows "z = y"
Borelsets_def: shows "\<bool> = sigma {S. \<exists>u. S = {..u}}"
compute_shift_map_keys: fixes t :: "'a"   and f :: "'c \<Rightarrow> 'b"   and xs :: "('a \<times> 'c) list" shows "punit.shift_map_keys t f (fmap_of_list xs) = fmap_of_list (map (\<lambda>(k, v). (punit.splus t k, f v)) xs)"
removeAllListDiff: fixes x :: "'b"   and a :: "'b list"   and b :: "'b list" shows "removeAll x (list_diff a b) = list_diff (removeAll x a) b"
assms_2: shows "H \<turnstile> A(i::=Zero)"
mset_set_Diff: fixes A :: "'e set"   and B :: "'e set" assumes "finite A"   and "B \<subseteq> A" shows "mset_set (A - B) = mset_set A - mset_set B"
simps_2: fixes xs :: "'a llist"   and n :: "nat" shows "lnth xs (Suc n) = (case xs of LNil \<Rightarrow> undefined (Suc n) | LCons x xs' \<Rightarrow> lnth xs' n)"
extended_Gromov_distance_nonneg: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "0 \<le> extended_Gromov_distance x y"
properfactorE: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "properfactor G a b" obtains "a divides\<^bsub>G\<^esub> b" and "\<not> b divides\<^bsub>G\<^esub> a"
balanced_Node_if_wbal2: fixes l :: "'a kdt"   and r :: "'a kdt"   and k :: "'a"   and v :: "real" assumes "balanced l"   and "balanced r"   and "\<bar>int (KD_Tree.size_kdt l) - int (KD_Tree.size_kdt r)\<bar> \<le> 1" shows "balanced (Node k v l r)"
size_eq_card_dom_lookup: fixes al :: "('a, 'b) alist" shows "size al = card (dom (DAList.lookup al))"
ide_map\<^sub>0_obj: fixes a :: "'a" assumes "C.obj a" shows "D.ide (G.map\<^sub>0 a)"
pax2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>(F \<longrightarrow> G) \<longrightarrow> \<circle>F \<longrightarrow> \<circle>G"