hcju/isabelleps huggingface dataset:
WT_gpv_right_gpv: fixes \<I>2 :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>1 :: "('d, 'e) \<I>" assumes "\<I>2 \<turnstile>g gpv \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<turnstile>g right_gpv gpv \<surd>"
coprime_diff_one_left: fixes a :: "'a" shows "coprime (a - (1::'a)) a"
mkKripke_simps_2: fixes ws :: "'c set"   and rels :: "'b \<Rightarrow> ('c \<times> 'c) set"   and val :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "relations (mkKripke ws rels val) = (\<lambda>a. Restr (rels a) ws)"
simps_7: fixes I :: "('b, 'c, 'a) interp"   and \<alpha> :: "('b, 'c, 'a) hp" shows "prog_sem I \<alpha>** = (prog_sem I \<alpha>)\<^sup>*"
Call1ThrowObj: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and M :: "String.literal"   and es :: "'addr expr1 list"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a\<bullet>M(es),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
M: assumes "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M" shows "Fun f T \<in> M"
comp_simp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.seq g f" shows "B.VV.comp g f = B.VxV.comp g f"
drlex_pm_trans: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "drlex_pm s t"   and "drlex_pm t u" shows "drlex_pm s u"
winning_region_deadends: fixes v :: "'a"   and p :: "Player" assumes "v \<in> VV p"   and "deadend v" shows "v \<in> winning_region p**"
one_over_poly_def: fixes p :: "nat \<Rightarrow> 'a" shows "one_over_poly p = (\<lambda>n. if n \<le> deg R p then p (deg R p - n) else \<zero>)"
ide_map\<^sub>0_obj: fixes a :: "'b" assumes "B.obj a" shows "C.ide (\<tau>\<^sub>0 a)"
interp_ins_maybe_grow3: fixes xs :: "('a \<times> 'a option) list"   and ys :: "('a \<times> 'a option) list" assumes "insert_ops (xs @ ys)" shows "\<exists>A\<subseteq>set (map fst ys). set (interp_ins (xs @ ys)) = set (interp_ins xs) \<union> A"
power_le_dvd: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat"   and b :: "'b"   and m :: "nat" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times (power.power one times a n) b"   and "m \<le> n" shows "dvd.dvd times (power.power one times a m) b"
imethds_def: fixes G :: "prog"   and I :: "qtname" shows "DeclConcepts.imethds G I = iface_rec G I (\<lambda>I i ts. Un_tables ts \<oplus>\<oplus> (set_option \<circ> table_of (map (\<lambda>(s, m). (s, I, m)) (imethods i))))"
f_f'_def_1: shows "f \<equiv> \<lambda>z. \<Sum>n. (a gchoose n) * z ^ n"
add2: fixes B :: "'k"   and k :: "'k"   and b :: "'k"   and a :: "'k" assumes "B \<equiv> k + b" shows "a + B \<equiv> k + (a + b)"
fim: shows "f ` \<Union> \<G> \<subseteq> rel_frontier T"
that: fixes u :: "'a list"   and v :: "'a list" assumes "u \<cdot> v \<cdot> u = w"   and "u \<noteq> \<epsilon>" shows "thesis"
oconf_blank: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and h :: "heap" assumes "P \<turnstile> C has_fields FDTs" shows "P,h \<turnstile> blank P C \<surd>"
h: fixes x :: "complex" assumes "x \<in> T" shows "exp (\<i> * complex_of_real (h x)) = (x - a) /\<^sub>R cmod (x - a) / ((x - b) /\<^sub>R cmod (x - b))"
cl4: fixes T :: "('a \<times> 'a set) set"   and R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" assumes "T \<parallel> T \<subseteq> T" shows "R \<cdot> T \<parallel> (S \<cdot> T) \<subseteq> R \<parallel> S \<cdot> T"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_vars_subset: fixes G :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s D"
cont_cfc: shows "continuous_on (S \<inter> {z. Im z \<le> 0}) (cnj \<circ> f \<circ> cnj)"
comm_mult_left_mono: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a \<le> b"   and "(0::'d) \<le> c" shows "c * a \<le> c * b"
rel_simps_69: shows "\<not> (1::'a) < (0::'a)"
frechet_correctness: fixes I :: "('a, 'b, 'c) interp"   and \<theta> :: "('a, 'c) trm"   and \<nu> :: "(real, 'c) vec" assumes "is_interp I"   and "dfree \<theta>" shows "(sterm_sem I \<theta> has_derivative frechet I \<theta> \<nu>) (at \<nu>)"
p_irred: shows "\<forall>a\<in>set P. case a of (E, k) \<Rightarrow> is_irreducible [C] E"
uminus_closed: fixes g :: "'g" assumes "g \<in> G" shows "- g \<in> G"
attribute_det: fixes classes :: "'a fset"   and associations :: "String.literal \<rightharpoonup>\<^sub>f String.literal \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and attributes :: "'a \<rightharpoonup>\<^sub>f String.literal \<rightharpoonup>\<^sub>f 'b"   and \<C> :: "'a"   and attr :: "String.literal"   and \<D>\<^sub>1 :: "'a"   and \<tau>\<^sub>1 :: "'b"   and \<D>\<^sub>2 :: "'a"   and \<tau>\<^sub>2 :: "'b" assumes "object_model classes associations"   and "unique_closest_attribute attributes \<C> attr \<D>\<^sub>1 \<tau>\<^sub>1"   and "unique_closest_attribute attributes \<C> attr \<D>\<^sub>2 \<tau>\<^sub>2" shows "\<D>\<^sub>1 = \<D>\<^sub>2 \<and> \<tau>\<^sub>1 = \<tau>\<^sub>2"
map_of_map_restrict: fixes f :: "'c \<Rightarrow> 'd"   and ks :: "'c list" shows "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
that: fixes a :: "'a"   and b :: "'a" assumes "cbox a b \<subseteq> A"   and "x \<in> box a b"   and "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i" shows "thesis"
supp_def: fixes x :: "'b" shows "supp x = {a. infinite {b. [(a, b)] \<bullet> x \<noteq> x}}"
face_cycle_succ_def: shows "H.face_cycle_succ \<equiv> edge_succ HM \<circ> edge_rev HM"
zero_V_def: shows "[]\<^sub>\<circ> \<equiv> set {}"
rep_eq: fixes x :: "'a" shows "(*\<^sub>V) (cblinfun_cinner_right x) = (\<bullet>\<^sub>C) x"
seqp_sos_subterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp"   and \<xi> :: "'f"   and a :: "'g seq_action"   and \<xi>' :: "'f"   and p' :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>"   and "\<exists>pn. p \<in> subterms (\<Gamma> pn)"   and "((\<xi>, p), a, \<xi>', p') \<in> seqp_sos \<Gamma>" shows "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
"1_2": shows "\<forall>k. \<chi> k = \<Phi> k * \<chi>\<^sub>1 k"
listrelp_imp_listsp1: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" assumes "listrelp (\<lambda>x y. P x) xs ys" shows "listsp P xs"
D_def: shows "D = (\<Prod>na\<leftarrow>n_as. if cmod (snd na) = 1 then d (snd na) else 1)"
list_ex_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "list_ex P xs = Bex (set xs) P"
distinct_9: fixes x21 :: "fmla"   and x22 :: "fmla"   and x41 :: "name"   and x42 :: "fmla" shows "x21 OR x22 \<noteq> exi x41 x42"
Abs_fmap_inject: fixes x :: "'a \<Rightarrow> 'b option"   and y :: "'a \<Rightarrow> 'b option" assumes "x \<in> {m. finite (dom m)}"   and "y \<in> {m. finite (dom m)}" shows "(map_of_fmap x = map_of_fmap y) = (x = y)"
transpose_of_unitary_is_unitary: fixes U :: "complex mat" assumes "unitary U" shows "unitary U\<^sup>t"
cardshift: fixes n :: "nat"   and n' :: "nat"   and p :: "nat \<Rightarrow> bool"   and n'' :: "nat" shows "card {i. n < i \<and> i \<le> n' \<and> p (n'' + i)} = card {i. n + n'' < i \<and> i \<le> n' + n'' \<and> p i}"
add_set_single1: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and x :: "'e"   and n :: "nat"   and y :: "'e" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "x \<in> add_set f {a} n"   and "y \<in> add_set f {a} n" shows "x = y"
black_def: fixes r :: "'g" shows "black r \<equiv> \<lambda>s. marked r s \<and> \<not> grey r s"
bij_betwE: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "\<forall>a\<in>A. f a \<in> B"
homeomorphic_simply_connected: fixes S :: "'b set"   and T :: "'c set" assumes "S homeomorphic T"   and "simply_connected S" shows "simply_connected T"
limsup_subseq_mono: fixes r :: "nat \<Rightarrow> nat"   and X :: "nat \<Rightarrow> 'b" assumes "strict_mono r" shows "limsup (X \<circ> r) \<le> limsup X"
bits_div_by_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide a zero = zero"
lookup_empty: shows "lookup Containers_Userguide.empty = Map.empty"
maxsimp_is_singleton: fixes x :: "'w set set" assumes "chamber x" shows "\<exists>w\<in>W. dual_order.poset_simplex_map \<P> {w} = x"
map_comp: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f1 :: "'j \<Rightarrow> 'g"   and f2 :: "'k \<Rightarrow> 'h"   and f3 :: "'l \<Rightarrow> 'i"   and v :: "('j, 'k, 'l) val" shows "map_val g1 g2 g3 (map_val f1 f2 f3 v) = map_val (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "b \<succeq> a" shows "b \<succ> a"
orderE: fixes b :: "'a upred"   and a :: "'a upred" assumes "a \<sqsubseteq> b" obtains "a = (a \<or> b)"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "Left_a.seq g f" shows "Left_a.cod (Left_a.comp g f) = Left_a.cod g"
map_transfer: fixes Rb :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Sd :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (rel_fun Rb Sd) (rel_fun (rel_resource' Rb) (rel_resource' Sd)) map_resource' map_resource'"
in_Edges_rev: fixes a :: "nat"   and b :: "nat"   and zs :: "nat list" shows "((a, b) \<in> Edges (rev zs)) = ((b, a) \<in> Edges zs)"
is_weakly_scdom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "is_weakly_scdom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> local.a_get_scdom_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<union> added_pointers \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
in_progress_dom: fixes r :: "'a \<Rightarrow> Runs.var set option"   and R :: "'a"   and x :: "Runs.var" assumes "in_progress (r R) x" shows "R \<in> dom r"
inverse_arrowsI: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "partial_magma.ide C (C g f)"   and "partial_magma.ide C (C f g)" shows "category.inverse_arrows C f g"
z_def2: shows "z = x + (1 - k) *\<^sub>R (y - x)"
intros_18: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e" shows "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> \<Rightarrow> Abrupt s"
returns_result_eq: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and y :: "'c"   and y' :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r y'" shows "y = y'"
msg_fresh_def: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" shows "msg_fresh \<sigma> m \<equiv> case m of Rreq hopsc x xa xb oipc osnc sipc \<Rightarrow> 1 \<le> osnc \<and> (sipc \<noteq> oipc \<longrightarrow> oipc \<in> kD (rt (\<sigma> sipc)) \<and> osnc \<le> nsqn (rt (\<sigma> sipc)) oipc \<and> (nsqn (rt (\<sigma> sipc)) oipc = osnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) oipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) oipc) = Aodv_Basic.inv)) | Rrep hopsc dipc dsnc x sipc \<Rightarrow> 1 \<le> dsnc \<and> (sipc \<noteq> dipc \<longrightarrow> dipc \<in> kD (rt (\<sigma> sipc)) \<and> dsnc \<le> nsqn (rt (\<sigma> sipc)) dipc \<and> (nsqn (rt (\<sigma> sipc)) dipc = dsnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) dipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) dipc) = Aodv_Basic.inv)) | Rerr destsc sipc \<Rightarrow> \<forall>ripc\<in>dom destsc. ripc \<in> kD (rt (\<sigma> sipc)) \<and> the (destsc ripc) - 1 \<le> nsqn (rt (\<sigma> sipc)) ripc | _ \<Rightarrow> True"
two_circles: fixes l :: "real"   and r :: "real"   and P :: "real poly"   and p :: "nat" assumes "l < r"   and "degree P \<le> p"   and "P \<noteq> 0"   and "p \<noteq> 0"   and "proots_count (map_poly complex_of_real P) (upper_circle l r \<union> lower_circle l r) = 1" shows "Bernstein_changes p l r P = 1"
padd_Cons_Cons: fixes h1 :: "'a"   and p1 :: "'a list"   and h2 :: "'a"   and p2 :: "'a list" shows "h1 # p1 +++ (h2 # p2) = (h1 + h2) # p1 +++ p2"
varFalse: fixes b :: "BExpr"   and t :: "Var \<Rightarrow> Val"   and s :: "Var \<Rightarrow> Val"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "\<not> evalB b t"   and " s \<approx> t " shows "(b, \<Phi>, s, t) \<in> var"
l_sub_smult: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K " shows "H \<subseteq> H \<diamondop> K"
proves_eq_elim_pre: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and B :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "term_ok \<Theta> B"   and "typ_of B = Some propT" shows "\<Theta>,{} \<turnstile> mk_eq A B \<longmapsto> A \<longmapsto> B"
prod_combine: fixes j :: "nat"   and i :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "j \<le> i" shows "f i * prod_list (map f [j..<i]) = prod_list (map f [j..<Suc i])"
parts_atomic: fixes x :: "msg" assumes "atomic x" shows "parts {x} = {x}"
minus_carrier_mat': fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A - B \<in> carrier_mat nr nc"
not_disjoint_union: fixes s :: "'b word"   and e :: "'b word"   and s' :: "'b word"   and e' :: "'b word" assumes "\<not> disjoint {s..e} {s'..e'}" shows "{s..e} \<union> {s'..e'} = {min s s'..max e e'}"
elims_2: fixes x :: "val"   and xa :: "val" assumes "x \<sqsubseteq> xa" obtains n :: "nat" and n' :: "nat" where "x = VNat n" and "xa = VNat n'" and "n = n'"   | f :: "(val \<times> val) fset" and f' :: "(val \<times> val) fset" where "x = val.Fun f" and "xa = val.Fun f'" and "fset f \<subseteq> fset f'"   | "x = Abs None" and "xa = Abs None"   | v :: "val" and v' :: "val" where "x = Abs (Some v)" and "xa = Abs (Some v')" and "v \<sqsubseteq> v'"   | "x = Wrong" and "xa = Wrong"
visits_via_ltl: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set" assumes "vmc_path_no_deadend G P v0 p \<sigma>"   and "visits_via P A W"   and "v0 \<notin> W" shows "visits_via (ltl P) A W"
foo25: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "((int \<times> int) set set, int) Combinators" assumes "wellformed_policy3 (p @ [x])" shows "wellformed_policy3 p"
bounded_clinear_compose: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd" assumes "bounded_clinear f"   and "bounded_clinear g" shows "bounded_clinear (\<lambda>x. f (g x))"
ith_conv: fixes x :: "'b"   and B :: "('c \<Rightarrow> 'b) set"   and i :: "'c" shows "(x \<in> ith B i) = (\<exists>f\<in>B. x = f i)"
locally_compatible_r_def: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset" shows "locally_compatible_r R1 R2 A B C \<equiv> \<forall>t u v. t \<in># A \<longrightarrow> u \<in># B \<longrightarrow> v \<in># C \<longrightarrow> (t, u) \<in> R2 \<longrightarrow> (u, v) \<in> R1 \<longrightarrow> (t, v) \<in> R2"
inner_scaleR_right: fixes x :: "'a"   and r :: "real"   and y :: "'a" shows "x \<bullet> r *\<^sub>R y = r * (x \<bullet> y)"
toSink_def: fixes s :: "'state" shows "toSink s \<equiv> SOME s'. s' \<in> \<delta> s \<inter> Sink"
run_sim1: fixes r :: "nat \<Rightarrow> 'u" assumes "is_run r" shows "G'.is_run (f \<circ> r)"
runit_simps_2: fixes f :: "'d" assumes "D.ide f" shows "src\<^sub>D \<r>\<^sub>D[f] = src\<^sub>D f"
ipurge_tr_rev_aux_append: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and xs :: "'c list"   and ys :: "'c list" shows "ipurge_tr_rev_aux I D U (xs @ ys) = ipurge_tr_rev_aux I D (sources_aux I D U ys) xs @ ipurge_tr_rev_aux I D U ys"
assms_3: fixes v :: "'al" assumes "v \<in> A" shows "u \<le>\<^sub>o\<^sub>w v"
ball_fun_contains_open_aux: fixes e :: "real"   and x :: "'a \<Rightarrow> 'b" assumes "0 < e" shows "\<exists>U. open U \<and> x \<in> U \<and> U \<subseteq> {y. dist x y < e}"
permute_minus_cancel_1: fixes p :: "perm"   and x :: "'c" shows "p \<bullet> - p \<bullet> x = x"
tinc_b_def: shows "tinc_b \<equiv> TMInc"
meet_def: fixes A :: "('b \<times> 'b) set" shows "A\<^sup>\<up> = (A\<inverse>)\<^sup>* O A\<^sup>*"
subst_fmla_subst_fmla: fixes x :: "name"   and y :: "name"   and u :: "trm"   and A :: "fmla"   and t :: "trm" assumes "x \<noteq> y"   and "atom x \<sharp> u" shows "(A(x::=t))(y::=u) = (A(y::=u))(x::=subst y u t)"
stutter_equiv_suffixes_left: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<sigma> \<approx> \<tau>" obtains m :: "nat" where "\<sigma> [m..] \<approx> \<tau> [n..]" and "\<forall>j<m. \<exists>k<n. \<sigma> [j..] \<approx> \<tau> [k..]"
restrict_neg: fixes x :: "'a option list" assumes "wf_tuple n C x"   and "P x" shows "list_all (\<lambda>A. P (restrict A x)) A_neg"
fun_relI: fixes A :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'g"   and f' :: "'f \<Rightarrow> 'h"   and B :: "('g \<times> 'h) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
monom: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "Q (monom P a n)"
norm_eq_norm_transpose: fixes A :: "'a sq_mtx" shows "\<parallel>A\<^sup>\<dagger>\<parallel> = \<parallel>A\<parallel>"
Int_open: fixes x :: "'a set"   and y :: "'a set" assumes "x open\<^bsub>T\<^esub>"   and "y open\<^bsub>T\<^esub>" shows "x \<inter> y open\<^bsub>T\<^esub>"
sel_6: fixes x21 :: "'a"   and x22 :: "'b"   and x23 :: "'a"   and x24 :: "'b" shows "new_id (Mult x21 x22 x23 x24) = x23"
filterlim_divide_at_bot_at_top_iff_2: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter"   and g :: "real \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> 0" shows "(LIM x F. f x / g x :> at_top) = ((g \<longlongrightarrow> 0) F \<and> (g has_sgnx sgn c) F)"
divides_ff_to_fract: fixes x :: "'b"   and y :: "'b" shows "divides_ff (to_fract x) (to_fract y) = (x dvd y)"
transition_cases_2: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b com \<times> 'b" assumes "(Basic f, s) -P1\<rightarrow> t" obtains "t = (Parallel [], f s)"
ContractsWithUnilateralSubstitutes_axioms: shows "ContractsWithUnilateralSubstitutes Xd Xh Pd Ch"
kat_4_equiv: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<cdot> y = d x \<cdot> y \<cdot> d z) = (d x \<cdot> y \<le> y \<cdot> d z)"
GF_advice_prop_entailment_1: fixes \<A> :: "'b ltln set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "\<A> \<Turnstile>\<^sub>P \<phi>[X]\<^sub>\<nu>" shows "{\<psi>. \<psi>[X]\<^sub>\<nu> \<in> \<A>} \<Turnstile>\<^sub>P \<phi>"
finite_Domain: fixes r :: "('c \<times> 'd) set" assumes "finite r" shows "finite (Domain r)"
integral_f: shows "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
DirProds_one': fixes i :: "'e"   and I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" assumes "i \<in> I" shows "\<one>\<^bsub>DirProds G I\<^esub> i = \<one>\<^bsub>G i\<^esub>"
xeq1_trans: fixes x :: "Value"   and s :: "Store"   and t :: "Store"   and r :: "Store" assumes "s \<equiv>[x] t"   and "t \<equiv>[x] r" shows "s \<equiv>[x] r"
proj2_rep_non_zero: fixes x :: "proj2" shows "proj2_rep x \<noteq> 0"
two_t_wise_eq_points_axioms: shows "two_t_wise_eq_points \<V> \<B> \<t> \<Lambda>\<^sub>t \<B>' \<Lambda>\<^sub>t'"
elims: fixes x :: "rat mat"   and xa :: "rat list list"   and xb :: "nat list list"   and xc :: "rat vec"   and y :: "rat mat \<times> nat list list \<times> rat list list" assumes "reduction_step x xa xb xc = y" obtains A :: "rat mat" and signs :: "rat list list" and subsets :: "nat list list" and lhs_vec :: "rat vec" where "x = A" and "xa = signs" and "xb = subsets" and "xc = lhs_vec" and "y = (let reduce_cols_A = reduce_mat_cols A lhs_vec; rows_keep = rows_to_keep reduce_cols_A in (take_rows_from_matrix reduce_cols_A rows_keep, take_indices subsets rows_keep, take_indices signs (find_nonzeros_from_input_vec lhs_vec)))"
exists_min_ground_head: fixes \<zeta> :: "('s, 'v) hd" shows "\<exists>f. f \<in> local.ground_heads \<zeta> \<and> (\<forall>g\<in>local.ground_heads \<zeta>. wt_sym f + \<delta>\<^sub>h * arity_sym\<^sub>h f \<le> wt_sym g + \<delta>\<^sub>h * arity_sym\<^sub>h g)"
exp_of_nat2_mult: fixes x :: "'a"   and n :: "nat" shows "exp (x * of_nat n) = exp x ^ n"
span_insert_idI: fixes r :: "'b \<Rightarrow>\<^sub>0 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "r \<in> phull B" shows "phull (insert r B) = phull B"
sgnx_poly_at_bot: fixes p :: "real poly" shows "sgnx (poly p) at_bot = sgn_neg_inf p"
complete_lattice_ccpo: shows "class.ccpo Sup (\<le>) (<)"
inducts: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv"   and n :: "enat" assumes "interaction_bounded_by consider gpv n" obtains "interaction_bound consider gpv \<le> n"
inter_add_left1: fixes x :: "'b"   and N :: "'b multiset"   and M :: "'b multiset" assumes "x \<notin># N" shows "add_mset x M \<inter># N = M \<inter># N"
case_distrib: fixes h :: "'d \<Rightarrow> 'b"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and prod :: "'e \<times> 'f" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"