hcju/isabelleps huggingface dataset:
infsetsum_reindex: fixes g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and f :: "'e \<Rightarrow> 'f" assumes "inj_on g A" shows "infsetsum f (g ` A) = (\<Sum>\<^sub>ax\<in>A. f (g x))"
class_exists_equiv: fixes cn :: "char list"   and P :: "'b prog" shows "(\<exists>x. fst x = cn \<and> x \<in> set P) = (class P cn \<noteq> None)"
less_is_drop_replicate: fixes x :: "'b word"   and n :: "nat" assumes "x < 2 ^ n" shows "to_bl x = replicate (LENGTH('b) - n) False @ drop (LENGTH('b) - n) (to_bl x)"
pure_the_pure: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "is_pure R" shows "pure (the_pure R) = R"
two_comparisons_into_case_order_6: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b"   and Q :: "'c"   and R :: "'c"   and P :: "'c" assumes "comparator comp" shows "(if lt_of_comp comp x y then Q else if lt_of_comp comp y x then R else P) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
filter_insort_triv: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" assumes "\<not> P x" shows "filter P (insort_key f x xs) = filter P xs"
dom_eqI: fixes a :: "'a1"   and f :: "'a1" assumes "C1.ide a"   and "C1.seq f a" shows "C1.dom f = a"
Nats_altdef2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "semiring_1.Nats one plus zero = {n \<in> ring_1.Ints one plus zero minus. less_eq zero n}"
dvd_0_left_iff: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times zero a = (a = zero)"
Diff_iff: fixes c :: "'f"   and A :: "'f set"   and B :: "'f set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
unique_connecting_path_properties_3: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "hd (v \<leadsto>\<^bsub>G\<^esub> w) = v"
inj_on_nth: fixes xs :: "'a list"   and I :: "nat set" assumes "distinct xs"   and "\<forall>i\<in>I. i < length xs" shows "inj_on ((!) xs) I"
min_less_iff_conj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "less z (ord.min less_eq x y) = (less z x \<and> less z y)"
rep_eq: fixes x :: "'a scene" shows "region x = {(s\<^sub>1, s\<^sub>2). \<forall>s. Rep_scene x s s\<^sub>1 = Rep_scene x s s\<^sub>2}"
prems_2: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" assumes "lowest_tops [i, t, e] = None"   and "i = Falseif" shows "P i t e"
abs_eq: shows "\<top> = Abs_clinear_space UNIV"
pr_simps_4: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "arr \<p>\<^sub>1[a0, a1]"
lub_eq: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'d \<Rightarrow> 'c"   and Y :: "'d \<Rightarrow> 'c" assumes "class.cpo below"   and "\<And>i. X i = Y i" shows "po.lub below (range X) = po.lub below (range Y)"
subgroup_indicator_pos: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := H'\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := H'\<rparr>)" shows "0 < subgroup_indicator (G\<lparr>carrier := H'\<rparr>) H a"
convert_term_id: fixes t :: "'b" assumes "no_abs t" shows "convert_term t = t"
norm_triangle_ineq4: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) \<le> norm a + norm b"
lm02: fixes X :: "'a set" shows "trivial X = (X = {} \<or> card X = 1)"
ovalidNF_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "ovalidNF P f Q \<equiv> \<forall>s. P s \<longrightarrow> f s \<noteq> None \<and> (\<forall>r. f s = Some r \<longrightarrow> Q r s)"
vintersection_vsingleton: fixes a :: "V" shows "a \<in>\<^sub>\<circ> set {a}"
frontier_translation_subtract: fixes a :: "'c"   and s :: "'c set" shows "frontier ((+) a ` s) = (+) a ` frontier s"
op_info_is_empty_def: shows "op_info_is_empty = is_empty"
path_image_subpath_gen: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b" shows "path_image (subpath u v g) = g ` closed_segment u v"
has_sgnx_const: fixes c :: "real"   and F :: "real filter" shows "((\<lambda>_. c) has_sgnx sgn c) F"
epm_1: fixes v :: "'b"   and w :: "'b"   and e :: "'b" assumes "bounded_distrib_lattice_class.comp_inf.vector v" shows "stone_algebra_class.comp_inf.prim_E w v e \<squnion> stone_algebra_class.comp_inf.prim_P w v e = stone_algebra_class.comp_inf.prim_EP w v e"
rel_advantage_def: fixes \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf" shows "okamoto_\<Sigma>_commit.rel_advantage \<A> \<equiv> spmf (okamoto_\<Sigma>_commit.rel_game \<A>) True"
authTicket_form: fixes A :: "agent"   and authK :: "nat"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV" shows "authK \<notin> range shrK \<and> authK \<in> symKeys \<and> authTicket = Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace>"
basisD_2: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set"   and x :: "'b" assumes "indep_system carrier indep"   and "indep_system.basis carrier indep X"   and "x \<in> carrier - X" shows "\<not> indep (insert x X)"
rel_interior_ball: fixes S :: "'c set" shows "rel_interior S = {x \<in> S. \<exists>e>0. ball x e \<inter> affine hull S \<subseteq> S}"
range_of_def: shows "range_of \<equiv> id \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst sas_plus_problem_ext_Tuple_Iso"
sorted_ins_list: fixes xs :: "'b list"   and x :: "'b" assumes "Sorted_Less.sorted xs" shows "Sorted_Less.sorted (ins_list x xs)"
ediv_def: fixes a :: "enat"   and k :: "nat" shows "ediv a k = (if a = \<infinity> then \<infinity> else enat (mydiv (THE i. a = enat i) k))"
normal_subgroup_axioms: shows "normal_subgroup L G (\<cdot>) \<one>"
Lim_transform_away_within: fixes a :: "'a"   and b :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "a \<noteq> b"   and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"   and "(f \<longlongrightarrow> l) (at a within S)" shows "(g \<longlongrightarrow> l) (at a within S)"
awalk_cost_append: fixes f :: "'b \<Rightarrow> real"   and xs :: "'b list"   and ys :: "'b list" shows "awalk_cost f (xs @ ys) = awalk_cost f xs + awalk_cost f ys"
neutral_const: fixes A :: "'c set" shows "(\<Sum>_\<in>A. (0::'a)) = (0::'a)"
well_typed: shows "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "idom_hom hom"   and "zero_hom_0 hom" shows "inj_idom_hom hom"
intros_13: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and k :: "i32"   and off :: "nat"   and t :: "t"   and vs :: "v list"   and a :: "nat" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "load m (nat_of_int k) off (t_length t) = None" shows "\<lparr>s;vs;[$C ConstInt32 k, $Load t None a off]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
final_def: fixes cfg :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate" shows "final cfg = (fst cfg = Skip \<or> fst cfg = Throw \<and> (\<exists>s. snd cfg = Normal s))"
inv_ide: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "C.VV.inv a = a"
seqI: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.arr f"   and "S.VV.arr g"   and "S.VV.dom g = S.VV.cod f" shows "S.VV.seq g f"
cong: fixes \<xi> :: "'c set" shows "rel_monoid.pid \<xi> \<equiv> rel_monoid.pid \<xi>"
least_def: fixes uu :: "'b itself"   and Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and S :: "('d, 'f) labeled_graph"   and G :: "('d, 'f) labeled_graph" shows "least uu Rs S G \<equiv> subgraph S G \<and> (\<forall>C. consequence_graph Rs C \<longrightarrow> maintained (S, G) C)"
is_nonterminal_cons_terminal: fixes x :: "symbol"   and k :: "nat"   and a :: "symbol list" assumes "is_terminal x"   and "k < length (x # a)"   and "is_nonterminal ((x # a) ! k)" shows "0 < k \<and> k - 1 < length a \<and> is_nonterminal (a ! (k - 1))"
composite_simps_1: shows "src (\<nu> \<star> \<pi>) = src \<pi>"
simps: fixes n :: "nat" shows "mopup n = mopup_a n @ shift mopup_b (2 * n)"
Tn_measure_preserving: fixes n :: "nat" shows "Tinv ^^ n \<in> measure_preserving M M"
comp_sinter_with: shows "local.comp_sinter_with = ord.sinter_with (lt_of_comp c)"
simps_2: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
coeff_in_ring2: fixes x :: "'d"   and A :: "'d set"   and a :: "'d \<Rightarrow> 'a" assumes "x \<in> A"   and "a \<in> A \<rightarrow> carrier R" shows "a x \<in> carrier R"
strict_mono_imp_inj: fixes f :: "'a \<Rightarrow> 'b" assumes "strict_mono f" shows "inj f"
accept: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and p :: "'d" assumes "Matching_Ternary.matches \<gamma> m action.Accept p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m action.Accept], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow"
map_diff_def: fixes f :: "'a \<Rightarrow> 'b option"   and ks :: "'a fset" shows "map_diff f ks = f |` (- fset ks)"
cases: fixes a1 :: "'a sequent \<times> nat"   and a2 :: "'a sequent list \<times> 'a sequent"   and a3 :: "('a sequent list \<times> 'a sequent) set" assumes "lastRule a1 a2 a3" obtains r :: "'a sequent list \<times> 'a sequent" and R :: "('a sequent list \<times> 'a sequent) set" and S :: "'a sequent" and \<Gamma> :: "'a form multiset" and \<Delta> :: "'a form multiset" where "a1 = ( \<Gamma>__ \<Rightarrow>* \<Delta>__, 0)" and "a2 = r__" and "a3 = R__" and "r__ \<in> Ax" and "Ax \<subseteq> R__" and "snd (extendRule S__ r__) = ( \<Gamma>__ \<Rightarrow>* \<Delta>__)"   | r :: "'a sequent list \<times> 'a sequent" and R :: "('a sequent list \<times> 'a sequent) set" and S :: "'a sequent" and \<Gamma> :: "'a form multiset" and \<Delta> :: "'a form multiset" and n :: "nat" where "a1 = ( \<Gamma> \<Rightarrow>* \<Delta>, n + 1)" and "a2 = r" and "a3 = R" and "r \<in> R" and "r \<notin> Ax" and "snd (extendRule S r) = ( \<Gamma> \<Rightarrow>* \<Delta>)" and "\<forall>p\<in>set (fst (extendRule S r)). \<exists>m\<le>n. (p, m) \<in> derivable R*"
src_vcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "seq \<nu> \<mu>" shows "src (\<nu> \<cdot> \<mu>) = src \<nu>"
minus: fixes ops :: "'c arith_ops_record"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "ring_ops ops R" shows "(R ===> R ===> R) (arith_ops_record.minus ops) (-)"
left_neutral: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and a :: "'c" assumes "monoid f z" shows "f z a = a"
ca_closed: fixes e :: "db" assumes "closed e" shows "evalDdb e\<cdot>env_empty_db \<triangleleft> e"
set_set_lvars: fixes l :: "lname \<Rightarrow> val option"   and l' :: "lname \<Rightarrow> val option"   and s :: "state" shows "set_lvars l (set_lvars l' s) = set_lvars l s"
assms: shows "\<nexists>!x. x \<in> U \<and> P x"
dfs_map_Some_idem: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and xs :: "('d, 'e) bintree list" shows "dfs f (map Some xs) = bts_dfs f xs"
1: shows "Some (\<omega>\<nu> o\<^sub>1) = d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2"
sees_method_is_class': fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "is_class P D"
refl_listrelp: fixes xs :: "'a list"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. R x x" shows "listrelp R xs xs"
lexord_irrefl: fixes R :: "('a \<times> 'a) set" assumes "irrefl R" shows "irrefl (lexord R)"
apply_sequence: fixes f :: "'a perm"   and g :: "'a perm"   and a :: "'a" shows "f \<langle>$\<rangle> (g \<langle>$\<rangle> a) = (f * g) \<langle>$\<rangle> a"
simps_14: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x11 :: "'a"   and x12 :: "'b" shows "Stmt.If x51 x52 x53 \<noteq> x11 \<leftarrow> x12"
sfilter_strict_scons: fixes p :: "'a \<rightarrow> tr"   and x :: "'a"   and xs :: "[:'a:]" assumes "p\<cdot>\<bottom> = \<bottom>" shows "sfilter\<cdot>p\<cdot>(x :# xs) = If p\<cdot>x then x :# sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs"
obj_fields_marked_good_ref_locs_hp_phaseD: fixes s :: "(('f, 'g, 'h) response, char list, 'e process_name, 'e process_name \<times> ('f, 'e, 'g, 'h) request_op, ('f, 'e, 'g, 'h) Model.local_state) system_state" assumes "atS gc obj_fields_marked_good_ref_locs s"   and "handshake_invL s" shows "sys_ghost_hs_phase s\<down> = hp_IdleMarkSweep \<and> All (ghost_hs_in_sync (s\<down> sys))"
fresh_1: fixes a :: "name"   and x1 :: "name" shows "a \<sharp> InputS x1 = a \<sharp> x1"
Compl_atMost: fixes k :: "'b" shows "- {..k} = {k<..}"
Bot_delR_simp: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "\<Gamma> \<Rightarrow> \<bottom>, \<Delta> = \<Gamma> \<Rightarrow> \<Delta>"
map_graph_selectors_1: fixes f :: "('d \<times> 'b) set"   and G :: "('c, 'd) labeled_graph" shows "vertices (map_graph f G) = f `` vertices G"
U_def: shows "U = arith_prog_fb (int_of_fbint x) d"
Table2_axioms: shows "Table2 f1 f2 f1' f2' e c l0 f1'' f2''"
Minimal_Solutions_alt_def: shows "local.Minimal_Solutions = {(x, y). (x, y) \<in> local.Solutions \<and> (x, y) \<noteq> (zeroes m, zeroes n) \<and> x \<le>\<^sub>v replicate m (Max (set b)) \<and> y \<le>\<^sub>v replicate n (Max (set a)) \<and> \<not> (\<exists>(u, v)\<in>local.Solutions. nonzero u \<and> u @ v <\<^sub>v x @ y)}"
bind_returns_heap_E: fixes h :: "'x"   and f :: "('x, 'y, 'aa) prog"   and g :: "'aa \<Rightarrow> ('x, 'y, 'z) prog"   and h'' :: "'x" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h''" obtains x :: "'aa" and h' :: "'x" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
o_bij: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "g \<circ> f = id"   and "f \<circ> g = id" shows "bij f"
weakOutputPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and M :: "'a"   and N :: "'a" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', P, Q) \<in> Rel" shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<leadsto><Rel> M\<langle>N\<rangle>.Q"
that: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>T. x \<in> T \<and> open T \<and> higher_differentiable_on T f n"
eq_commute: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (b = a)"
simps_3: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "formula_of_constraint (t1, True, MLessEq, t2) = formula.LessEq t1 t2"
normalize_simps_3: fixes r :: "('d \<times> 'd) set" shows "Language.normalize (Spec r) = Spec r"
dom_def: fixes f :: "(bool, bool list) concrete_category.arr" shows "local.dom f = (if domains f \<noteq> {} then SOME a. a \<in> domains f else null)"
basic_subtype_x_Integer: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B Integer" obtains "\<tau> = UnlimitedNatural"
simps: fixes q :: "'b queue" shows "is_empty q = (list q = [])"
vector_scalar_commute: fixes A :: "(('b, 'd) vec, 'c) vec"   and c :: "'b"   and x :: "('b, 'd) vec" shows "A *v (c *s x) = c *s (A *v x)"
intro: fixes hom :: "'b \<Rightarrow> 'a" assumes "hom (0::'b) = (0::'a)" shows "zero_hom hom"
transitionI_2: fixes P :: "pi"   and P''' :: "pi"   and a :: "name"   and x :: "name"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'''"   and "P''' \<longmapsto> a<\<nu>x> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"
join': fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a"   and b :: "'a" assumes "merge_on A h bo m"   and "a \<in> A" shows "\<forall>ab. (m a b = Some ab) = (bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u))"
axioms_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "semilattice f" shows "abel_semigroup f"
lift_subst_dummy: fixes s :: "dB"   and i :: "nat"   and dummy :: "dB" assumes "\<not> free s i" shows "lift (s[dummy/i]) i = s"
tp_add: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "trigpoly f"   and "trigpoly g" shows "trigpoly (\<lambda>x. f x + g x)"
det_id: fixes P :: "'a process" shows "(P [+] P) = P"
some_geodesic_is_geodesic_segment_2: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.geodesic_space dist uniformity open" shows "geodesic_segment {x--y}"
sims: fixes as :: "'a"   and cs :: "'c"   and l :: "'l"   and cs' :: "'c" assumes "A.reachable as"   and "C.reachable cs"   and "R as cs"   and "clstep l cs cs'" shows "\<exists>as'. R as' cs' \<and> alstep l as as'"
leadsTo_UN: fixes I :: "'d set"   and F :: "'e program"   and A :: "'d \<Rightarrow> 'e set"   and B :: "'e set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i \<longmapsto> B" shows "F \<in> \<Union> (A ` I) \<longmapsto> B"
\<tau>diverge_mbisim_inv: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2" shows "r1.mthr.\<tau>diverge s1 = r2.mthr.\<tau>diverge s2"
index_swaps_last_size: fixes s :: "'b list"   and sws :: "nat list" assumes "distinct s" shows "length s \<le> index (swaps sws s) (last s) + length sws + 1"
MP_null: fixes A :: "fmla"   and B :: "fmla"   and H :: "fmla set" assumes "{} \<turnstile> A IMP B"   and "H \<turnstile> A" shows "H \<turnstile> B"
M_def: shows "M \<equiv> {x \<in> X\<^sub>0. \<forall>y\<in>X\<^sub>0. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r}"
Nil_subset_T: fixes P :: "'b process" shows "{[]} \<subseteq> T P"
mat_inv_cnj: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_cnj (mat_inv M) = mat_inv (mat_cnj M)"
collinear_3_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "collinear {a, b, c}"   and "collinear {b, c, d}"   and "b \<noteq> c" shows "collinear {a, b, d}"
SC_trans: fixes X :: "'p set"   and Y :: "'p set"   and Z :: "'p set" assumes "SC X Y"   and "SC Y Z" shows "SC X Z"
finite: fixes m :: "('a, 'b) RBT.rbt" assumes "rm.invar m" shows "finite (dom (rm.\<alpha> m))"
S6_live: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> $ImpInv rmhist p) \<and> SF(MClkReply memCh crCh cst p)_c p \<and> \<box>\<diamond>S6 rmhist p \<longrightarrow> \<box>\<diamond>S1 rmhist p"
fun_relD2: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'f"   and Ra :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'f) set"   and x' :: "'c"   and r' :: "'f" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f' x' = r'" shows "\<forall>x. (x, x') \<in> Ra \<longrightarrow> (f x, r') \<in> Rr"
vsv_vdomain_vrange_vsingleton: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "\<D>\<^sub>\<circ> r = ZFC_in_HOL.set {a}"   and "\<R>\<^sub>\<circ> r = ZFC_in_HOL.set {b}" shows "r = ZFC_in_HOL.set {\<langle>a, b\<rangle>}"
cmh: shows "continuous_map (prod_topology (top_of_set {0..1}) (subtopology (nsphere n) {x. - 1 / 2 \<le> x k})) (nsphere n) h"
o_inv_distrib: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd" assumes "bij f"   and "bij g" shows "inv (f \<circ> g) = inv g \<circ> inv f"
totient_eq: fixes p :: "nat" assumes "residues_prime p" shows "totient p = p - 1"
finite_field_factorization_modulo_ring: fixes g :: "'a mod_ring poly"   and f :: "int poly"   and d :: "'a mod_ring"   and gs :: "'a mod_ring poly list"   and c :: "int"   and fs :: "int poly list" assumes "g = of_int_poly f"   and "square_free_m f"   and "finite_field_factorization g = (d, gs)"   and "c = to_int_mod_ring d"   and "fs = map to_int_poly gs" shows "unique_factorization_m f (c, mset fs)"
translation_def: shows "translation = (\<lambda>a\<in>G. restrict ((\<cdot>) a) G)"
Vaxiom_def: shows "Vaxiom \<equiv> $\<phi> vid1 local.empty \<rightarrow> [[$\<alpha> vid1]]$\<phi> vid1 local.empty"
assms_2: shows "- z \<notin> path_image \<gamma>"
distinct_33: fixes x21 :: "'d"   and x22 :: "'a \<Rightarrow> 'a"   and x23 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x21}\<lbrakk>x22\<rbrakk> x23 \<noteq> call(x10)"
lm045: fixes g1 :: "'d \<Rightarrow> 'c list list"   and G :: "'d"   and g2 :: "'c list \<Rightarrow> 'f \<Rightarrow> 'e list"   and N :: "'f"   and f2 :: "'c set \<Rightarrow> 'f \<Rightarrow> 'e set" shows "(\<forall>l\<in>set (g1 G). set (g2 l N) = f2 (set l) N) \<longrightarrow> {f2 P N |P. P \<in> set (map set (g1 G))} = set (map (\<lambda>l. set (g2 l N)) (g1 G))"
select_convs_2: fixes root :: "'c"   and edges :: "'c edge set"   and more :: "'d" shows "edges \<lparr>root = root, edges = edges, \<dots> = more\<rparr> = edges"
word_split: fixes v :: "nat \<Rightarrow> 'a"   and k :: "nat" obtains v\<^sub>1 :: "'a list" and v\<^sub>2 :: "nat \<Rightarrow> 'a" where "v = v\<^sub>1 \<frown> v\<^sub>2" and "length v\<^sub>1 = k"
list_induct2: fixes xs :: "'a list"   and ys :: "'b list"   and P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
inj_hom_dimension: fixes K :: "'a list set"   and E :: "'a list set"   and n :: "nat" assumes "subfield K (K [X])"   and "\<one>\<^bsub>Rupt K p\<^esub> \<noteq> \<zero>\<^bsub>Rupt K p\<^esub>"   and "inj_on (rupture_surj K p) E"   and "UP.dimension n K E" shows "Hom.dimension n (rupture_surj K p ` K) (rupture_surj K p ` E)"
f0: shows "gate 3 (Quantum.Id 2 \<Otimes> X)"
assms: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r\<lparr>a\<rparr> = f a"
u_2: fixes x :: "'a" assumes "x \<in> A" shows "u x = 1"