hcju/isabelleps huggingface dataset:
normalize_prod_mset_normalize: fixes A :: "'a multiset" shows "normalize (\<Prod>\<^sub># (image_mset normalize A)) = normalize (\<Prod>\<^sub># A)"
WT_converter_id: fixes \<I> :: "('a, 'b) \<I>" shows "\<I>, \<I> \<turnstile>\<^sub>C id_converter \<surd>"
the_Class_def: fixes T :: "ty" shows "the_Class T = (SOME C. T = Class C)"
ImZ_of_nat: fixes n :: "nat" shows "ImZ (of_nat n) = 0"
f2: shows "\<b> = int (size \<P>) * (\<v> div \<k>)"
non_empty_cycle_root_equivalences_3: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "non_empty_cycle_root r x"   and "point q" shows "(r \<le> x\<^sup>\<star> ; q) = (x\<^sup>T ; q \<noteq> 0)"
wfp_on_subset: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<subseteq> B"   and "wfp_on P B" shows "wfp_on P A"
iT_Plus_neg_iSince_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<S> t2 I \<oplus>- k. Q t2) = (P (t1 - k). t1 \<S> t2 I \<down>\<ge> k. Q (t2 - k))"
mono_D2: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P [+] S) \<longrightarrow> Ra (P [+] S) s = Ra (Q [+] S) s"
l2_inv7_skr: fixes R :: "fid_t"   and K :: "msg" shows "{l2_inv7 \<inter> l2_inv6} l2_skr R K {> l2_inv7}"
iFROM_cut_le1: fixes t :: "nat"   and n :: "nat" assumes "t \<in> [n\<dots>]" shows "[n\<dots>] \<down>\<le> t = [n\<dots>,t - n]"
simps: fixes v1 :: "'d"   and p1 :: "'c"   and v2 :: "'d"   and p2 :: "'b" shows "edge_end ((v1, p1), v2, p2) = v2"
dbl_inc_simps_3: fixes minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and uminus :: "'f \<Rightarrow> 'f"   and one :: "'f" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one one = numeral.numeral one plus (num.Bit1 num.One)"
exhaust: fixes y :: "('e, 'd) Term.term" obtains x1 :: "'d" where "y = Var x1"   | x21 :: "'e" and x22 :: "('e, 'd) Term.term list" where "y = Fun x21 x22"
is_span: shows "C.span trg.leg0 trg.leg1"
nact_exists: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<and> \<not> (\<exists>k\<ge>n. latestAct_cond c t i k)"
rel_interior_projection: fixes S :: "('b \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c set"   and y :: "'b"   and z :: "'c" assumes "convex S"   and "f = (\<lambda>y. {z. (y, z) \<in> S})" shows "((y, z) \<in> rel_interior S) = (y \<in> rel_interior {y. f y \<noteq> {}} \<and> z \<in> rel_interior (f y))"
wfzf_minimal: fixes R :: "(ZF \<times> ZF) set"   and C :: "ZF set" assumes "wfzf R"   and "C \<noteq> {}" shows "\<exists>x. x \<in> C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> C)"
divide_gt: fixes b :: "'a"   and c :: "rat"   and a :: "'a" assumes "b < c *R a"   and "0 < c" shows "(1 / c) *R b < a"
assms_3: shows "xs \<noteq> ys \<or> x \<noteq> y"
Abs_G_perm_diff: fixes g :: "'a"   and h :: "'a" assumes "g \<in> G"   and "h \<in> G" shows "\<pp> g - \<pp> h = \<pp> (binop g (\<ii> h))"
Atm_WbisT: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and atm :: "'b"   and tval :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "PL_Indis indis"   and "PL_Indis.compatAtm aval indis atm" shows "(Atm atm, Atm atm) \<in> PL_Indis.WbisT tval aval indis"
B: shows "ereal (- dist x y) \<le> Sup {limsup (\<lambda>n. ereal (dist x (u n) - dist y (u n))) |u. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi}"
runit_in_vhom: fixes f :: "'a"   and a :: "'a" assumes "ide f"   and "src f = a" shows "\<guillemotleft>\<r>[f] : f \<star> a \<Rightarrow> f\<guillemotright>"
hm_lookup_def: fixes k :: "'b"   and hm :: "('b, 'a) hashmap" shows "hm_lookup k hm \<equiv> HashMap_Impl.lookup k (impl_of_RBT_HM hm)"
renaming_preserves_tautology: fixes C :: "'at Literal set"   and Sel :: "'at set" assumes "tautology C" shows "tautology (rename_clause Sel C)"
vsv_vdomain_vrange_vsingleton: fixes a :: "V"   and b :: "V" assumes "\<D>\<^sub>\<circ> A = ZFC_in_HOL.set {a}"   and "\<R>\<^sub>\<circ> A = ZFC_in_HOL.set {b}" shows "A = ZFC_in_HOL.set {\<langle>a, b\<rangle>}"
comp_L_2: shows "\<lbrakk>\<lbrakk>t\<^sub>0\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk> = Maps.MkArr (src (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) (src t) (Maps.Comp \<lbrakk>t\<^sub>0\<rbrakk> \<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>)"
overlap_def: fixes x :: "'b ivl"   and y :: "'b ivl" shows "overlap x y = (low y \<le> high x \<and> low x \<le> high y)"
dg_is_arrD_4: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "\<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"
q_def: shows "q \<equiv> poly_of_vec (vec_first (vec n c) (n - d))"
Ioi_le_Ico: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.preorder less_eq less" shows "ord.greaterThan less a \<subseteq> ord.atLeast less_eq a"
that: fixes g :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P \<and> f = g \<oplus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f) \<and> deg R g < deg R f" shows "thesis"
CFG_Construct_wf_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set" shows "CFG_Construct_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> CFG_Construct \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses"
join_trees: fixes g\<^sub>1 :: "'a ugraph"   and g\<^sub>2 :: "'a ugraph"   and u :: "'a"   and v :: "'a" assumes "tree g\<^sub>1"   and "tree g\<^sub>2"   and "nodes g\<^sub>1 \<inter> nodes g\<^sub>2 = {}"   and "u \<in> nodes g\<^sub>1"   and "v \<in> nodes g\<^sub>2" shows "tree (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2))"
composite_is_arrow: shows "hseq (\<nu> \<star> \<pi>) \<rho>"
dist_triangle2: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "dist x y \<le> dist x z + dist y z"
order_def: fixes S :: "('a, 'b) monoid_scheme" shows "Coset.order S = card (carrier S)"
Diff_Compl: fixes A :: "'d set"   and B :: "'d set" shows "A - - B = A \<inter> B"
x'eq: shows "x = (\<Sum>t\<in>insert a T'. c' t *\<^sub>R t)"
inj_combinator: fixes x :: "'a"   and S :: "'a set"   and T :: "'a \<Rightarrow> 'b set" assumes "x \<notin> S" shows "inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
functor_axioms: shows "functor J (\<cdot>\<^sub>C) E"
is_character_data_ptr_kind_def: fixes ptr :: "(_) node_ptr" shows "is_character_data_ptr_kind ptr = (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> True)"
iterates_above_le_f: fixes x :: "'a"   and a :: "'a" assumes "x \<in> iterates_above a"   and "a \<in> Field leq" shows "(x, f x) \<in> leq"
some: shows "wordinterval_CIDR_split1 rs = (Some x2, y)"
all_total_def: fixes F :: "'c program" shows "all_total F \<equiv> \<forall>act\<in>Acts F. Domain act = UNIV"
max_subset_def: fixes A :: "'b set"   and P :: "'b set \<Rightarrow> bool" shows "max_subset A P = (ARG_MAX card B. B \<subseteq> A \<and> P B)"
col_one_side: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "Col A B C"   and "A \<noteq> C"   and "A B OS P Q" shows "A C OS P Q"
arg_cong2: fixes a :: "'b"   and b :: "'b"   and c :: "'c"   and d :: "'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "a = b"   and "c = d" shows "f a c = f b d"
\<tau>exec_moves_conv_\<tau>exec_meth: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list" shows "\<tau>exec_moves ci P t es = \<tau>exec_meth ci (compP2 P) (compEs2 es) (compxEs2 es 0 0) t"
finite: shows "finite (R\<^sup>* `` \<Sigma>i)"
mono_Int: fixes f :: "'b set \<Rightarrow> 'c set"   and A :: "'b set"   and B :: "'b set" assumes "mono f" shows "f (A \<inter> B) \<subseteq> f A \<inter> f B"
supp_of_finite_sets: fixes S :: "'a set" assumes "finite S" shows "supp S = \<Union> (supp ` S)"
not_QuadRes_ex_if_prime: fixes p :: "int" assumes "prime p"   and "odd p" shows "\<exists>x. \<not> QuadRes p x"
Issues_def: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" shows "A Issues B with X on evs = (\<exists>Y. Says A B Y \<in> set evs \<and> X \<in> parts {Y} \<and> X \<notin> parts (knows Spy (takeWhile (\<lambda>z. z \<noteq> Says A B Y) (rev evs))))"
sorted_append: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (xs @ ys) = (linorder.sorted less_eq xs \<and> linorder.sorted less_eq ys \<and> (\<forall>x\<in>ran\<^sub>l xs. \<forall>y\<in>ran\<^sub>l ys. less_eq x y))"
assms_7: shows "\<forall>x. x \<in> ChSetQ \<longrightarrow> x \<in> loc PQ"
rcosI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "h \<otimes> x \<in> H #> x"
diff_assign_lem1: fixes I :: "('sf, 'a, 'sz) interp"   and i :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "dterm_sem I (if i = vid1 then $' vid1 else Const 0) (fst \<nu>, \<chi>y. if vid1 = y then Functions I fid1 (\<chi>i. dterm_sem I (local.empty i) \<nu>) else snd \<nu> $ y) = dterm_sem I (if i = vid1 then $f fid1 local.empty else Const 0) \<nu>"
slp_Append: fixes n :: "'node"   and as :: "'edge list"   and n'' :: "'node"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^bsub>sl\<^esub>* n''"   and "n'' -as'\<rightarrow>\<^bsub>sl\<^esub>* n'" shows "n -as @ as'\<rightarrow>\<^bsub>sl\<^esub>* n'"
nodes_le_def: fixes xs :: "nat list"   and ys :: "nat list" shows "nodes_le xs ys \<equiv> Sorted_Less.sorted (rev ys) \<and> Sorted_Less.sorted (rev xs) \<and> set ys \<subseteq> set xs \<or> xs = ys"
hom_component_lpp_2: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "lpp (hom_component p (deg_pm (lpp p))) = lpp p"
leI: fixes x :: "'b Word.word"   and y :: "'b Word.word" assumes "\<not> x <s y" shows "y \<le>s x"
inter_union_distrib_right: fixes C :: "'k multiset"   and A :: "'k multiset"   and B :: "'k multiset" shows "C + A \<inter># B = (C + A) \<inter># (C + B)"
Quotient_transaction: shows "Quotient (=) transaction_merkle.Transaction' transaction_merkle.the_Transaction transaction_merkle.cr_transaction"
rel_of_pred_def: fixes s :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_of_pred s \<equiv> {(a, b). s a b}"
closed_segment_commute: fixes a :: "'b"   and b :: "'b" shows "{a--b} = {b--a}"
ideal_generated_dvd_eq_diff_set: fixes i :: "'b"   and I :: "'b set"   and J :: "'b set" assumes "i \<in> I"   and "i \<notin> J"   and "\<forall>j\<in>J. i dvd j"   and "finite J" shows "ideal_generated I = ideal_generated (I - J)"
TimesR_def: fixes R :: "'d regex set"   and s :: "'d regex" shows "TimesR R s = (\<lambda>r. Times r s) ` R"
matches_comply_exact: fixes \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and p :: "'d"   and m :: "'c match_expr"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "action" assumes "ternary_ternary_eval (map_match_tac \<beta> p m) \<noteq> TernaryUnknown"   and "matcher_agree_on_exact_matches \<gamma> \<beta>" shows "Semantics.matches \<gamma> m p = Matching_Ternary.matches (\<beta>, \<alpha>) m a p"
islimpt_UNIV: fixes x :: "'a" shows "x islimpt UNIV"
Cons_1: fixes Qs :: "('c multiset \<Rightarrow> 'a) list" assumes "list_all carrier_coeff Ps"   and "list_all (\<lambda>P. index_free P i) Ps"   and "list_all carrier_coeff Qs"   and "list_all (\<lambda>Q. index_free Q i) Qs"   and "indexed_eval_aux Ps i = indexed_eval_aux Qs i"   and "length Ps = length Qs" shows "Ps = Qs"
singleton_bot: fixes default :: "unit \<Rightarrow> 'a" shows "singleton default \<bottom> = default ()"
Gen_def: fixes P :: "(symbol \<times> character list) list set" shows "Gen P = {uu_. \<exists>x p. uu_ = x \<and> p \<in> P \<and> pvalid p x}"
det_row_uminus: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and a :: "('c, 'd) vec" shows "Square_Matrix.det (upd_row M i (- a)) = - Square_Matrix.det (upd_row M i a)"
r_def: shows "r \<equiv> a *\<^sub>R p + (1 - a) *\<^sub>R q"
rep_eq: fixes x :: "'c pmf" shows "pmf x = (\<lambda>x. measure_pmf.prob x {x})"
sd_efficient_sds_axioms_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "sd_efficient_sds_axioms agents alts sds \<equiv> \<forall>R. pref_profile_wf agents alts R \<longrightarrow> SD_efficient R (sds R)"
that: fixes cs :: "'b fset" assumes "(fst (f a), cs) |\<in>| fgroup_by f as"   and "snd (f a) |\<in>| cs" shows "thesis"
us: shows "us = getFrN (map snd txs) (map fst txs) [cnj \<phi>1 \<phi>2] (length txs)"
le_norm_blinfun: fixes f :: "'b \<Rightarrow>\<^sub>L 'a"   and x :: "'b" shows "norm (blinfun_apply f x) / norm x \<le> norm f"
assms_2: shows "g1 \<in> stabilizer G \<phi> x"
lookup_update_some: fixes env :: "('f, 'g, 'h) env"   and xs :: "'h list"   and e :: "('f, 'g, 'h) env"   and env' :: "('f, 'g, 'h) env" assumes "lookup env xs = Some e" shows "lookup (update xs (Some env') env) xs = Some env'"
pos_le_divideR_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "real"   and a :: "'b"   and b :: "'b" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less_eq a (scaleR (inverse c) b) = less_eq (scaleR c a) b"
map_eq_imp_length_eq: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and g :: "'e \<Rightarrow> 'c"   and ys :: "'e list" assumes "map f xs = map g ys" shows "length xs = length ys"
type_less_x_ObjectType: fixes \<tau> :: "'a basic_type"   and \<D> :: "'a" assumes "\<tau> < \<langle>\<D>\<rangle>\<^sub>\<T>" obtains "\<tau> = OclVoid"   | \<C> :: "'a" where "\<tau> = \<langle>\<C>\<rangle>\<^sub>\<T>" and "\<C> < \<D>"
adverse_at_01_2: fixes s :: "nat \<Rightarrow> nat option"   and z :: "nat"   and i :: "nat"   and j :: "nat" assumes "s \<in> \<R>" shows "adverse s z i j 1 \<down>= j"
filter_a_def: shows "filter_a = filter (\<lambda>x. x = a)"
hball_One: fixes P :: "hf \<Rightarrow> bool" shows "HBall 1 P = P 0"
extremeD_1: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and e :: "'c" assumes "extreme X r e" shows "e \<in> X"
rel_map_1: fixes Sb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e \<Rightarrow> 'c"   and x :: "'e pmf"   and y :: "'d pmf" shows "rel_pmf Sb (map_pmf i x) y = rel_pmf (\<lambda>x. Sb (i x)) x y"
ide_char: fixes a :: "'b \<times> 'b \<times> 'b" shows "D.VVV.ide a = (D.VVV.arr a \<and> D.VxVxV.ide a)"
is_red_subset: fixes A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_red A p"   and "A \<subseteq> B" shows "punit.is_red B p"
internal_int_poly_factorization: fixes f :: "int poly"   and c :: "int"   and fs :: "(int poly \<times> nat) list" assumes "coeff f 0 \<noteq> 0"   and "local.internal_int_poly_factorization f = (c, fs)" shows "square_free_factorization f (c, fs)"
simps_15: fixes x4 :: "32 word" shows "Intg x4 \<noteq> Null"
OUT_h'_A: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "d_OUT h' x = d_OUT (h 0) x + g (\<langle>x\<rangle>, vertex.SINK) - g (vertex.SINK, \<langle>x\<rangle>)"
get_fstn_args_take: fixes xs :: "nat list"   and m :: "nat"   and n :: "nat" assumes "length xs = m"   and "n \<le> m" shows "map (\<lambda>f. rec_exec f xs) (get_fstn_args m n) = take n xs"
Var: fixes l :: "char list \<Rightarrow> val option"   and V :: "char list"   and v :: "val"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "l V = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>Var V,(h, l)\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"
arcE: fixes e :: "'a \<times> 'a"   and u :: "'a"   and v :: "'a" assumes "G.arc e (u, v)" obtains "e \<in> parcs G" and "fst e = u" and "snd e = v"
characters_def: fixes G :: "('a, 'b) monoid_scheme" shows "characters G = {\<chi>. character G \<chi>}"
VUnionE: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> A" obtains x :: "V" where "a \<in>\<^sub>\<circ> x" and "x \<in>\<^sub>\<circ> A"
mHom_eq: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i"   and g :: "'e \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N"   and "\<forall>m\<in>carrier M. f m = g m" shows "f = g"
comp_def: fixes g :: "'a"   and f :: "'a" shows "g \<cdot>\<^sup>o\<^sup>p f \<equiv> f \<cdot> g"
elts_imp_ground: fixes u :: "tm"   and t :: "tm"   and S :: "atom set" assumes "u \<in> elts t"   and "ground_aux t S" shows "ground_aux u S"
rotate_comp_eq: fixes w :: "'b list"   and n :: "nat" assumes "w \<bowtie> rotate n w" shows "rotate n w = w"
sdom_def: fixes m :: "'c \<Rightarrow> 'd set" shows "sdom m = {a. m a \<noteq> {}}"
fls_lr_inverse_one_one_2: shows "fls_right_inverse 1 (1::'a) = 1"
intros_1: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> r" shows "(a, b) \<in> r\<^sup>+"
eq_Min_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(m = local.Min A) = (m \<in> A \<and> (\<forall>a\<in>A. le m a))"
reachable_def: fixes s :: "'s"   and s' :: "'s" shows "reachable s s' \<equiv> \<exists>p. local.path s p s'"
card_UN_b: fixes K :: "nat set"   and j0 :: "nat"   and j :: "nat" assumes "K \<subseteq> {j0<..}"   and "finite K"   and "j0 \<le> card K"   and "j \<le> j0" shows "card (\<Union>i<j. b (enum K i) (j0, i)) = enum (d j0) j - enum (d j0) 0"
existence_ivl_zero: fixes x0 :: "'c" assumes "x0 \<in> Csafe" shows "0 \<in> existence_ivl0 x0"
"1_3": shows "Ind {a} (set w\<^sub>1)"
simps_10: fixes t :: "'a E.term"   and u :: "'a E.term"   and v :: "'a E.term" shows "\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<rbrace> = \<alpha>' (\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>)"
almost_all_commutative': fixes S :: "'b set"   and P :: "'b \<Rightarrow> nat \<Rightarrow> bool" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> \<forall>\<^sub>\<infinity>i. P x i" shows "\<forall>\<^sub>\<infinity>i. \<forall>x\<in>S. P x i"
lift_SDG: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list"   and nx :: "'h"   and H :: "'i set"   and L :: "'i set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "SDG src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main NewExit (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) (lift_ParamDefs ParamDefs) (lift_ParamUses ParamUses)"
FV0_def: fixes uu :: "unit"   and fm :: "presb" shows "FV0 uu fm = (case fm of presb.Eq is x xa \<Rightarrow> {n. n < length is \<and> is ! n \<noteq> 0})"
eq_defined_reflD_2: fixes a :: "'a" shows "(TT = eq\<cdot>a\<cdot>a) = (a \<noteq> \<bottom>)"
S_def: shows "S \<equiv> sylvester_mat_sub m n p q"
vimage_mono: fixes A :: "'c set"   and B :: "'c set"   and f :: "'d \<Rightarrow> 'c" assumes "A \<subseteq> B" shows "f -` A \<subseteq> f -` B"
conj: fixes a :: "complex"   and b :: "complex" shows "conjugate a * b = conjugate (a * conjugate b)"
minimal: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.pcpo below" shows "below (pcpo.bottom below) x"
strict_subtuple_rtranclp_intro: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<rightharpoonup>\<^sub>f 'b \<Rightarrow> 'b"   and ym :: "'a \<rightharpoonup>\<^sub>f 'b"   and xm :: "'a \<rightharpoonup>\<^sub>f 'b" assumes "\<And>xm ym. R (f xm) (f ym) \<Longrightarrow> strict_subtuple (\<lambda>x y. R x y \<or> x = y) xm ym"   and "bij_on_trancl R f"   and "acyclicP_on (fmran' ym) R"   and "R\<^sup>+\<^sup>+ (f xm) (f ym)" shows "strict_subtuple R\<^sup>*\<^sup>* xm ym"
star_slide: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<star> \<cdot> x = x \<cdot> (y \<cdot> x)\<^sup>\<star>"
Compl_disjoint2: fixes A :: "'d set" shows "- A \<inter> A = {}"
ord_isom_less: fixes E :: "('g, 'h) Order_scheme"   and f :: "'a \<Rightarrow> 'g"   and a :: "'a"   and b :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec> b = f a \<prec>\<^bsub>E\<^esub> f b"
convexD_alt: fixes s :: "'a set"   and a :: "'a"   and b :: "'a"   and u :: "real" assumes "convex s"   and "a \<in> s"   and "b \<in> s"   and "0 \<le> u"   and "u \<le> 1" shows "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> s"
fresh_1: fixes a :: "name"   and x1 :: "char list" shows "a \<sharp> TVar x1 = a \<sharp> x1"
inverse_monoid_isomorphism: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" assumes "monoid_isomorphism \<eta> M composition unit M' composition' unit'" shows "monoid_isomorphism (restrict (inv_into M \<eta>) M') M' composition' unit' M composition unit"
stake_invert_Nil: fixes n :: "nat"   and s :: "'d stream" shows "(stake n s = []) = (n = 0)"
Match_2: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and b :: "name"   and x :: "name"   and P' :: "pi"   and a :: "name" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>b<x> \<prec> P'" shows "[a\<frown>a]P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>b<x> \<prec> P'"
0: shows "f\<^sup>2 - 4 * v \<noteq> 0"
pp_inf_p: fixes x :: "'a" shows "- - x \<sqinter> - x = bot"
eps_def: shows "eps = (\<lambda>i. 1 / real (i + 1))"
simps_1: fixes f :: "'a \<Rightarrow> 'a" shows "iterate 0 f = id"
assms: shows "P \<rightarrow> \<langle>\<alpha>,Q\<rangle>"
x: fixes y :: "'a" assumes "y \<in> S" shows "orthogonal x y"
w\<^sub>g'_is_map: shows "is_left_adjoint w\<^sub>g'"