hcju/isabelleps huggingface dataset:
deformation_retraction_relative_homology_group_isomorphisms: fixes X :: "'a topology"   and Y :: "'b topology"   and r :: "'a \<Rightarrow> 'b"   and s :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and V :: "'b set"   and p :: "int" assumes "retraction_maps X Y r s"   and "r ` U \<subseteq> V"   and "s ` V \<subseteq> U"   and "homotopic_with (\<lambda>h. h ` U \<subseteq> U) X X (s \<circ> r) id" shows "group_isomorphisms (relative_homology_group p X U) (relative_homology_group p Y V) (hom_induced p X U Y V r) (hom_induced p Y V X U s)"
pcompose_power_left: fixes p :: "'a poly"   and n :: "nat"   and q :: "'a poly" shows "p ^ n \<circ>\<^sub>p q = (p \<circ>\<^sub>p q) ^ n"
Abs_poly_inverse: fixes y :: "nat \<Rightarrow> 'b" assumes "y \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'b)}" shows "poly.coeff (Abs_poly y) = y"
stateful_strand_step_fv_subset_cases_7: fixes X :: "'b list"   and F :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list"   and G :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list"   and S :: "('a, 'b) stateful_strand_step list" assumes "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G - set X \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
rep_eq: fixes x :: "'x fset" shows "sorted_list_of_fset x = sorted_list_of_set (fset x)"
PUpdate_DefaultPUpdate: shows "PUpdate (\<lambda>D. DefaultPData (Data.DataSpace D))"
fresh_vsubst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X" shows "fresh zs z (X #[y1 // y]_ys) = ((zs = ys \<and> z = y \<or> fresh zs z X) \<and> (fresh ys y X \<or> zs \<noteq> ys \<or> z \<noteq> y1))"
rpr_less_imp_le: fixes x :: "'c"   and r :: "('c \<times> 'c) set"   and y :: "'c" assumes "x \<^bsub>r\<^esub>\<prec> y" shows "x \<^bsub>r\<^esub>\<preceq> y"
isom_gch_units_transpTr2: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "i \<le> n"   and "j \<le> n"   and "i < j"   and "g i \<cong> E" shows "{i. i \<le> n \<and> g i \<cong> E} = insert i ({i. i \<le> n \<and> g i \<cong> E} - {i})"
canE_unitor_1: fixes f :: "'a E.term" assumes "E.Ide f" shows "B.can f (f \<^bold>\<star> E.Src f) = B.runit (EVAL f)"
normal_vector_span: fixes D :: "'b vec set"   and n :: "nat" assumes "Suc (card D) = n"   and "D \<subseteq> carrier_vec n"   and "finite D"   and "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) D" shows "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) D = {x \<in> carrier_vec n. x \<bullet> gram_schmidt.normal_vector n D = (0::'b)}"
PO_hoare_def: fixes pre :: "'e set"   and R :: "('e \<times> 'e) set"   and post :: "'e set" shows "{pre} R {> post} \<equiv> R `` pre \<subseteq> post"
carrier_adjunct_ord: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D" shows "carrier (adjunct_ord D a) = carrier D \<union> {a}"
intro: fixes f\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "cancel_semigroup f\<^sub>a"   and "cancel_semigroup f\<^sub>b" shows "cancel_semigroup_pair f\<^sub>a f\<^sub>b"
ltl_Until_Release_con_1: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> R\<^sub>c \<psi> = (\<not> \<xi> \<Turnstile>\<^sub>c not\<^sub>c \<phi> U\<^sub>c not\<^sub>c \<psi>)"
row_of_matrix_def: fixes A :: "'b matrix"   and m :: "nat" shows "row_of_matrix A m \<equiv> take_rows (move_matrix A (- int m) 0) 1"
ahm_update_correct_1: fixes hm :: "('a, 'b) ArrayHashMap_Impl.hashmap"   and k :: "'a"   and v :: "'b" assumes "ArrayHashMap_Impl.ahm_invar hm" shows "ArrayHashMap_Impl.ahm_invar (ArrayHashMap_Impl.ahm_update k v hm)"
simps_1: fixes uu :: "('e \<times> 'e) set"   and uv :: "'f \<Rightarrow> 'e"   and u :: "'e" shows "c_sources uu uv u [] = {u}"
nprv_ptrmE_uni0: fixes \<sigma> :: "'fmla"   and n :: "nat"   and F :: "'fmla set"   and t :: "'trm"   and \<psi> :: "'fmla" assumes "\<sigma> \<in> ptrm n"   and "nprv F \<sigma>"   and "nprv F (subst \<sigma> t out)"   and "nprv (insert (eql (Var out) t) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t \<in> trm" shows "nprv F \<psi>"
maps_to_nI: fixes f :: "nat \<Rightarrow> int"   and k :: "int" assumes "\<And>n. f n \<in> {0..k}" shows "maps_to_n k f"
poly_mult: fixes p :: "'c poly"   and q :: "'c poly"   and x :: "'c" shows "poly (p * q) x = poly p x * poly q x"
fim: shows "f ` topspace (nsphere (n - Suc 0)) \<subseteq> topspace (nsphere (n - Suc 0))"
assms_6: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp" assumes "P1 scrut"   and "P1 e1"   and "P1 e2" shows "P1 (scrut ? e1 : e2)"
dom_dom: fixes f :: "'s" assumes "arr f" shows "local.dom (local.dom f) = local.dom f"
simps_3: fixes v0 :: "'a"   and rs :: "('a, 'b) result list" shows "sequence_result (Rval v0 # rs) = map_result ((#) v0) id (sequence_result rs)"
case_4: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> ((nat \<Rightarrow> nat option) \<Rightarrow> bool) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'a"   and f4 :: "'a"   and f5 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'a" shows "(case Fence of Read x xa xb \<Rightarrow> f1 x xa xb | Write x xa xb xc xd xe xf \<Rightarrow> f2 x xa xb xc xd xe xf | RMW x xa xb xc xd xe xf xg xh \<Rightarrow> f3 x xa xb xc xd xe xf xg xh | Fence \<Rightarrow> f4 | Ghost x xa xb xc \<Rightarrow> f5 x xa xb xc) = f4"
strip_while_coeffs: fixes p :: "'b poly" shows "strip_while ((=) (0::'b)) (coeffs p) = coeffs p"
wuntil_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "wuntil \<phi> \<psi> \<equiv> dis (until \<phi> \<psi>) (alw \<phi>)"
n_circ_5: fixes k :: "nat" shows "path_image (n_circ k) = (if k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
prems_1: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : [] _> ts"
that: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g \<in> set (Macaulay_list ps)"   and "g \<noteq> 0"   and "lt p = lt g" shows "thesis"
not_dvd_pderiv: fixes p :: "'c poly" assumes "degree p \<noteq> 0" shows "\<not> p dvd pderiv p"
oc: shows "open_cover_of_open_subset U it.ind_is_open V I W"
lookup_mult_scalar_tt_tt: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "lookup (p \<odot> q) (punit.tt p \<oplus> tt q) = punit.tc p * tc q"
cardSuc_mono_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Card_order r"   and "Card_order r'" shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
Cons_Cons: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" assumes "xs = x # y # ys" shows "thesis"
Id_in_Acts: fixes F :: "'b program" shows "Id \<in> Acts F"
finite_measure_def: fixes M :: "'a measure" shows "finite_measure M \<equiv> sigma_finite_measure M \<and> finite_measure_axioms M"
support_Un: fixes z :: "'e"   and A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "support z (A \<union> B) f = support z A f \<union> support z B f"
iT_Minus_0: fixes k :: "nat" shows "k \<ominus> {0} = {k}"
terminated_no_commands: fixes p :: "'g"   and sh :: "('i, 'h, 'g, 'j, 'k) system_state" assumes "atLs p {{}} sh" shows "\<exists>s. GST sh p = ([], s)"
resultant_impl_rec_def: shows "resultant_impl_rec = subresultant_prs_main_impl (\<lambda>(Gk, hk). if degree Gk = 0 then hk else (0::'b))"
tendsto_divide_ereal: fixes f :: "'b \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal"   and m :: "ereal" assumes "(f \<longlongrightarrow> l) F"   and "(g \<longlongrightarrow> m) F"   and "m \<noteq> 0"   and "\<not> (\<bar>l\<bar> = \<infinity> \<and> \<bar>m\<bar> = \<infinity>)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> l / m) F"
eta_cases_1: fixes T :: "typ"   and s :: "Core.term"   and z :: "Core.term" assumes "Abs T s \<rightarrow>\<^sub>\<eta> z" obtains sa :: "Core.term" where "s = sa $ Bv 0" and "z = decr 0 sa" and "\<not> is_dependent sa"   | t :: "Core.term" where "z = Abs T t" and "s \<rightarrow>\<^sub>\<eta> t"
igOpIPresIGWlsSTR_imp_igOpIPresIGWls: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igOpIPresIGWlsSTR MOD" shows "igOpIPresIGWls MOD"
g1: fixes id :: "'id" assumes "\<parallel>id\<parallel>\<^bsub>k\<^esub>"   and "\<parallel>id\<parallel>\<^bsub>k\<^esub>" shows "id = the_singleton"
add: fixes A :: "(('a, 'c) vec, 'b) vec"   and b1 :: "('a, 'c) vec"   and b2 :: "('a, 'c) vec" shows "A *v (b1 + b2) = A *v b1 + A *v b2"
ttree_restr_nxt_subset2: fixes x :: "'b"   and S :: "'b set"   and t :: "'b ttree" assumes "x \<notin> S" shows "paths (ttree_restr S (nxt t x)) \<subseteq> paths (ttree_restr S t)"
intros_2: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and u :: "'f \<Rightarrow> 'g"   and d :: "'g"   and l' :: "'e"   and u' :: "'f \<Rightarrow> 'g" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsup>d\<^esup> \<langle>l', u'\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"
eqvt_def: fixes f :: "'c" shows "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"
Union: fixes X :: "'a set set set"   and A :: "'a set set" assumes "X \<subseteq> subset.suc_Union_closed A" shows "\<Union> X \<in> subset.suc_Union_closed A"
is_RB_inD: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and u :: "'t"   and g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "G \<subseteq> dgrad_sig_set' (length fs) d"   and "is_RB_in d rword G u"   and "\<not> is_syz_sig d u"   and "d (pp_of_term u) \<le> dgrad_max d"   and "is_canon_rewriter rword G u g" shows "rep_list g \<noteq> 0"
assms: fixes i :: "'n" shows "w $h i \<le> v $h i"
path_inv_Cons: fixes a :: "bool"   and ds :: "bool list"   and T :: "tree" shows "path (a # ds) T \<longrightarrow> (\<exists>l r. T = Branching l r)"
simps_3: fixes e :: "(name, hf) finfun"   and t :: "tm"   and u :: "tm" shows "\<lbrakk>Eats t u\<rbrakk>e = \<lbrakk>t\<rbrakk>e \<triangleleft> \<lbrakk>u\<rbrakk>e"
sum_pos2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and I :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite I"   and "i \<in> I"   and "less zero (f i)"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq zero (f i)" shows "less zero (comm_monoid_add.sum plus zero f I)"
lv_relI: fixes x :: "real list" assumes "length x = DIM('a)" shows "(x, eucl_of_list x) \<in> lv_rel"
one_mat_ge_zero: fixes n :: "nat" shows "1\<^sub>m n \<ge>\<^sub>m 0\<^sub>m n n"
closure_of_closed: fixes S :: "('d, 'e) eq_object_scheme"   and A :: "'d set" shows "closure_of\<^bsub>S\<^esub> A \<subseteq> carrier S"
add_le_less_mono: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset"   and d :: "'c multiset" assumes "a \<subseteq># b"   and "c \<subset># d" shows "a + c \<subset># b + d"
distinct_5: fixes x2 :: "'b"   and x31 :: "'b trm"   and x32 :: "'b trm" shows "Const x2 \<noteq> x31 \<cdot> x32"
rotate_trailE: fixes u :: "'a"   and p :: "'b list"   and w :: "'a" assumes "trail u p u"   and "w \<in> set (awalk_verts u p)" obtains q :: "'b list" and r :: "'b list" where "p = q @ r" and "trail w (r @ q) w" and "set (awalk_verts w (r @ q)) = set (awalk_verts u p)"
lin: fixes x :: "'b"   and y :: "'b" shows "R (h x) (h y) \<or> R (h y) (h x)"
sum_list_ennreal: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> f x" shows "(\<Sum>x\<leftarrow>xs. ennreal (f x)) = ennreal (sum_list (map f xs))"
assms_1: shows "u \<turnstile>\<^bsub>v,n\<^esub> M1"
k': shows "(spr_simAbs (fst k'), snd k') \<in> (\<Union>a. SPRdet.sim_equiv_class a ` SPR.jkbpC) \<times> UNIV"
execute_operator_sas_plus_def: fixes s :: "'a \<Rightarrow> 'b option"   and op :: "('a, 'b) sas_plus_operator" shows "s \<then>\<^sub>+ op \<equiv> s ++ map_of (effect_of op)"
monic_irreducible_gcd: fixes f :: "'b poly"   and u :: "'b poly" assumes "monic f"   and "irreducible f" shows "gcd f u \<in> {1, f}"
IH: fixes accu :: "('a \<times> 'a) list" assumes "\<forall>a\<in>set M. case a of (a, b) \<Rightarrow> SecurityInvariant_complies_formal_def a b"   and "wf_list_graph G"   and "set Es \<subseteq> set (edgesL G)"   and "set accu \<subseteq> set (edgesL G)"   and "distinct (Es @ accu)" shows "TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G (get_impl M) accu Es = TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu (list_graph_to_graph G) (get_spec M) accu Es"
reduce_nested_mpoly_0: shows "reduce_nested_mpoly 0 = 0"
hyps_2: fixes S :: "'node SDG_node set" assumes "n'' \<in> set ns"   and "nx \<in> S" shows "n'' \<in> HRB_slice S"
list_all_init: shows "list_all (\<lambda>q. q \<in> reach (set \<Sigma>) \<delta> q\<^sub>0) [q\<^sub>0]"
multiplicative_function'_axioms_def: fixes f :: "nat \<Rightarrow> 'b"   and f_prime_power :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and f_prime :: "nat \<Rightarrow> 'b" shows "multiplicative_function'_axioms f f_prime_power f_prime \<equiv> (\<forall>p k. prime p \<longrightarrow> 0 < k \<longrightarrow> f (p ^ k) = f_prime_power p k) \<and> (\<forall>p. prime p \<longrightarrow> f_prime_power p 1 = f_prime p)"
assms: shows "\<forall>x\<in>X. f x = x"
2: shows "g = ZFfun (|dom|g) (|cod|g) ((|@|) g)"
prems_4: shows "P \<turnstile> Ts' [\<le>] (T # Ts)"
is_sig_red_top_addsE: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg (=) F p" obtains f :: "'t \<Rightarrow>\<^sub>0 'b" where "f \<in> F" and "rep_list f \<noteq> 0" and "rep_list p \<noteq> 0" and "punit.lt (rep_list f) adds punit.lt (rep_list p)" and "sing_reg (punit.lt (rep_list p) \<oplus> lt f) (punit.lt (rep_list f) \<oplus> lt p)"
par_comm: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "times a b = times b a"
union_empty: fixes A :: "'s" shows "A \<^bold>\<union> empty = A"
finite_UnI: fixes F :: "'d set"   and G :: "'d set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
sign_power: fixes x :: "'a"   and n :: "nat" shows "sign (x ^ n) = (if n = 0 then 1 else if even n then \<bar>sign x\<bar> else sign x)"
totalValuationForFormulaDefinesItsValue: fixes valuation :: "Literal list"   and formula :: "Literal list list" assumes "CNF.total valuation (vars formula)" shows "formulaTrue formula valuation \<or> formulaFalse formula valuation"
conf_ClassD: fixes P :: "'a prog"   and h :: "heap"   and v :: "val"   and C :: "char list" assumes "P,h \<turnstile> v :\<le> Class C" shows "v = Null \<or> (\<exists>a obj T. v = Addr a \<and> h a = \<lfloor>obj\<rfloor> \<and> obj_ty obj = T \<and> subtype P T (Class C))"
resPres: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "P \<sim> Q" shows "<\<nu>x>P \<sim> <\<nu>x>Q"
comp_converter_id_right: fixes conv :: "('h, 'i, 'j, 'k) converter" shows "conv \<odot> 1\<^sub>C = conv"
span_minimal: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "m1.subspace T" shows "m1.span S \<subseteq> T"
empty_def: shows "Abs_Int2_ivl.empty = {1\<dots>0}"
intros_1: fixes x :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)"   and y :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)" assumes "x = y" shows "A.neval x 0 y"
nat_sub2: fixes b :: "nat"   and a :: "nat"   and d :: "nat"   and c :: "nat" assumes "b \<le> a"   and "d \<le> c"   and "c - d \<le> a - b" shows "a - b - (c - d) = a + d - (b + c) \<and> b + c \<le> a + d"
plusU_assoc': fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c"   and returnU :: "udom \<rightarrow> udom\<cdot>'c"   and bindU :: "udom\<cdot>'c \<rightarrow> (udom \<rightarrow> udom\<cdot>'c) \<rightarrow> udom\<cdot>'c"   and plusU :: "udom\<cdot>'c \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c"   and a :: "udom\<cdot>'c"   and b :: "udom\<cdot>'c"   and c :: "udom\<cdot>'c" assumes "class.monad_plus fmapU returnU bindU plusU" shows "plusU\<cdot>(plusU\<cdot>a\<cdot>b)\<cdot>c = plusU\<cdot>a\<cdot>(plusU\<cdot>b\<cdot>c)"
rk2_fas_err_def: fixes rkp :: "floatarith"   and x0 :: "floatarith list"   and h :: "floatarith"   and cx :: "floatarith list"   and s2 :: "floatarith" shows "rk2_fas_err rkp x0 h cx s2 = map (rk2_fas_err_nth rkp x0 h cx s2) [0..<length x0]"
nc_discovered_eq_finished: assumes "\<not> cond s"   and "\<not> is_break param s" shows "dom (discovered s) = dom (finished s)"
UEQ: fixes e :: "real" assumes "0 < e" shows "\<exists>d>0. \<forall>n. \<forall>x\<in>S. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (\<F> n x') (\<F> n x) < e"
resChainFresh: fixes x :: "name"   and xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "x \<sharp> \<lparr>\<nu>*xvec\<rparr>P = (x mem xvec \<or> x \<sharp> P)"
conv_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linorder_stone_relation_algebra_expansion one times uminus less_eq less bot top inf sup conv" shows "conv x = x"
Min_antimono: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c set"   and N :: "'c set" assumes "class.linorder less_eq less"   and "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "less_eq (linorder.Min less_eq N) (linorder.Min less_eq M)"
full_Suc_Branch2_iff: fixes n :: "nat"   and l :: "'a tree23"   and p :: "int \<times> 'a"   and r :: "'a tree23" shows "full (Suc n) (Branch2 l p r) = (full n l \<and> full n r)"
pairwise_rel_prime: shows "pairwise_coprime_list qs"
ring_irreducibleI': fixes R :: "('c, 'd) ring_scheme"   and r :: "'c" assumes "domain R"   and "r \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}"   and "irreducible (Ring_Divisibility.mult_of R) r" shows "ring_irreducible\<^bsub>R\<^esub> r"
h3: shows "\<turnstile> P \<and> Enabled \<langle>M\<rangle>_g \<longrightarrow> Enabled \<langle>A\<rangle>_f"
assms: fixes x :: "'a"   and y :: "'a" assumes "P x y" shows "Q x y"
globally_consistent_adapt_bisim: fixes c\<^sub>1 :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "'Com"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "globally_consistent A mds" shows "\<langle>c\<^sub>1, mds, mem\<^sub>1 [\<parallel>\<^sub>1 A]\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2 [\<parallel>\<^sub>2 A]\<rangle>"
measurable_snd': fixes f :: "'c \<Rightarrow> 'd \<times> 'e"   and M :: "'c measure"   and N :: "'d measure"   and P :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M N \<Otimes>\<^sub>M P" shows "(\<lambda>x. snd (f x)) \<in> M \<rightarrow>\<^sub>M P"
dyadics_def: shows "dyadics \<equiv> \<Union>k m. {of_nat m / (2::'b) ^ k}"
standard_def: fixes C :: "('b \<times> 'c) set"   and b :: "'b"   and t :: "'b"   and idt :: "'b"   and G :: "('b, 'c) labeled_graph" shows "standard C b t idt G \<equiv> graph G \<and> vertices G \<noteq> {} \<and> ident_rel idt G \<and> getRel b G = {} \<and> getRel t G = {(x, y). x \<in> vertices G \<and> y \<in> vertices G} \<and> (\<forall>(l, v)\<in>C. getRel l G = {(v, v)})"
is_span: shows "C.span \<nu>.dom.leg0 \<nu>.dom.leg1"
Bad_inter: fixes U :: "'a set"   and n :: "nat" shows "Bad U n = (T ^^ n) --` U \<inter> Bad0 n"
show_vec_real_def: fixes v :: "(real, 'b) vec" shows "show_vec_real v = (\<chi>i. show_real (v $ i))"
PTrans: shows "\<tau>.P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P'"
DisjointPaths_extend: fixes P :: "'a list" assumes "v0 \<leadsto>P\<leadsto>\<^bsub>H_x\<^esub> v1"   and "\<And>xs v. \<lbrakk>xs \<in> Q - {Q_hit}; xs \<noteq> P; v \<in> set xs; v \<in> set P\<rbrakk> \<Longrightarrow> v = v0 \<or> v = v1" shows "DisjointPaths H_x v0 v1 (insert P (Q - {Q_hit}))"
lubIa: fixes cl :: "'b potype"   and S :: "'b set"   and L :: "'b" assumes "CL cl"   and "S \<subseteq> pset cl"   and "isLub S cl L" shows "L = lub S cl"
bal_i_delete: fixes t :: "'a tree"   and dl :: "nat"   and x :: "'a"   and t' :: "'a tree"   and dl' :: "nat" assumes "bal_i (size t + dl) (height t)"   and "local.delete x (t, dl) = (t', dl')" shows "bal_i (size t' + dl') (height t')"
elims: fixes x :: "ty"   and y :: "String.literal option" assumes "class_type_of' x = y" obtains C :: "String.literal" where "x = Class C" and "y = \<lfloor>C\<rfloor>"   | T :: "ty" where "x = T\<lfloor>\<rceil>" and "y = \<lfloor>Object\<rfloor>"   | "x = Void" and "y = None"   | "x = Boolean" and "y = None"   | "x = Integer" and "y = None"   | "x = NT" and "y = None"