hcju/isabelleps huggingface dataset:
tr_init: fixes c :: "('p, 's, 'm) configuration" shows "trace c [] c"
subtype_narrow: fixes \<Delta> :: "binding list"   and X :: "tyvrs"   and Q :: "ty"   and \<Gamma> :: "binding list"   and M :: "ty"   and N :: "ty"   and P :: "ty" assumes "(\<Delta> @ [TVarB X Q] @ \<Gamma>)\<turnstile>M<:N"   and "\<Gamma>\<turnstile>P<:Q" shows "(\<Delta> @ [TVarB X P] @ \<Gamma>)\<turnstile>M<:N"
union_preserves_multiset: fixes M :: "'k \<Rightarrow> nat"   and N :: "'k \<Rightarrow> nat" assumes "M \<in> multiset"   and "N \<in> multiset" shows "(\<lambda>a. M a + N a) \<in> multiset"
assms: fixes y :: "xml" assumes "y \<in> set B" shows "monotone sum_bot.le_fun sum_bot_ord (C y)"
HEndPhase2_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
transp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n"   and "j < m" shows "p A i j"
assoc: fixes a :: "'e multiset"   and b :: "'e multiset"   and c :: "'e multiset" shows "a \<union># b \<union># c = a \<union># (b \<union># c)"
poly_of_vars_def: fixes \<phi> :: "char list \<Rightarrow> nat"   and xs :: "char list multiset" assumes "poly_embed \<phi>" shows "poly_embed.poly_of_vars \<phi> xs = fold_mset (\<lambda>a. (*) (Var (\<phi> a))) 1 xs"
renaming_distr_store: fixes \<beta> :: "'a \<Rightarrow> 'a"   and \<alpha> :: "'b \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and l :: "'a"   and v :: "('b, 'a, 'c) val" assumes "bij \<beta>" shows "\<R>\<^sub>S \<alpha> \<beta> (\<sigma>(l \<mapsto> v)) = \<R>\<^sub>S \<alpha> \<beta> \<sigma>(\<beta> l \<mapsto> \<R>\<^sub>V \<alpha> \<beta> v)"
IH: fixes I :: "'variable \<Rightarrow> 'domain option" assumes "\<forall>op\<in>set \<psi>. are_operator_effects_consistent op op"   and "G \<subseteq>\<^sub>m execute_serial_plan_sas_plus I \<psi>" shows "G \<subseteq>\<^sub>m execute_parallel_plan_sas_plus I (List_Supplement.embed \<psi>)"
4: shows "v \<sqinter> - r = t\<^sup>T\<^sup>\<star> * r \<sqinter> - r"
zero_infty_def: shows "0 \<equiv> Num (0::'b)"
basis_for_obtain_unique_scalars: fixes vs :: "'v list"   and v :: "'v" assumes "set vs \<subseteq> V \<and> V = RSpan vs \<and> R_lin_independent vs"   and "v \<in> V" shows "\<exists>!as. length as = length vs \<and> v = as \<bullet>\<cdot> vs"
assms_2: shows "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"
path_component_def: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" shows "path_component S x y \<equiv> \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
sum0: shows "(\<Sum>x\<in>X. \<beta> x * exp x) = 0"
lex_polychain_of_last: fixes c :: "real \<times> real"   and d :: "real \<times> real"   and x0 :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "(c, d) \<in> set (polychain_of x0 xs)"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "Counterclockwise_2D_Arbitrary.lex (snd (last (polychain_of x0 xs))) d"
funcsetI: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'g"   and B :: "'g set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> A \<rightarrow> B"
vecd: fixes x :: "real" assumes "x \<in> {a..b}" shows "(f has_vector_derivative f' x) (at x within {a..b})"
FV_EqButGH: fixes p :: "pvar"   and p' :: "pvar" shows "FV (EqButGH p p') \<subseteq> {p, p'}"
list_slice_eq_list_slice_take: fixes xs :: "'b list"   and k :: "nat" shows "list_slice (xs \<down> (length xs div k * k)) k = list_slice xs k"
clear_locals_def: fixes \<xi> :: "state" shows "clear_locals \<xi> = \<xi>\<lparr>msg := SOME x. True, data := SOME x. True, dests := SOME x. True, pre := SOME x. True, rreqid := SOME x. True, dip := SOME x. True, oip := SOME x. True, hops := SOME x. True, dsn := SOME x. True, dsk := SOME x. True, osn := SOME x. True, sip := SOME x. x \<noteq> ip \<xi>\<rparr>"
pcr_cr_eq: shows "pcr_key_order (=) = cr_key_order"
R27_bc_2: shows "pmf (sds R27) c = 0"
sparse_row_vector_minus: fixes v :: "(nat \<times> 'a) list" shows "sparse_row_vector (minus_spvec v) = - sparse_row_vector v"
calc_ideal_point_1_unit: fixes A :: "complex"   and B :: "complex"   and z1 :: "complex"   and z2 :: "complex" assumes "is_real A"   and "(cmod A)\<^sup>2 < (cmod B)\<^sup>2"   and "(z1, z2) = calc_ideal_point1_cvec A B" shows "z1 * cnj z1 = z2 * cnj z2"
forall_elim': 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 \<Gamma> :: "Core.term set"   and \<tau> :: "typ"   and B :: "Core.term"   and a :: "Core.term" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ B"   and "\<turnstile>\<^sub>\<tau> a : \<tau>"   and "wf_term (sig \<Theta>) a"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> B \<bullet> a"
assms: shows "\<exists>subs. raw_match_term t2 t1 Map.empty = Some subs"
action_repc: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "action (repc cfg) = rept (reps (state cfg)) (action cfg)"
simps_2: fixes r :: "real"   and x :: "quat" shows "Im1 (r *\<^sub>R x) = r * Im1 x"
oconf_is_type: fixes G :: "prog"   and s :: "st"   and obj :: "obj"   and a :: "loc" assumes "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>Inl a" shows "is_type G (obj_ty obj)"
hrange_def: fixes r :: "hf" shows "hrange r = hdomain (hconverse r)"
is_hashtable_def: fixes l :: "('c \<times> 'd) list list"   and ht :: "('c, 'd) hashtable" shows "is_hashtable l ht = the_array ht \<mapsto>\<^sub>a l * \<up> (ht_size l (the_size ht) \<and> ht_hash l \<and> ht_distinct l \<and> 1 < length l)"
assms_2: fixes I :: "'b"   and a :: "'a"   and b :: "'a"   and J :: "'b"   and c :: "'a"   and d :: "'a" assumes "Q I a b"   and "Q J c d"   and "[[a c b d]]" shows "P I J"
units_of_carrier: fixes G :: "('c, 'd) monoid_scheme" shows "carrier (units_of G) = Units G"
simps_1: fixes x :: "('idx, 'pred, 'act) formula" shows "tau_steps x 0 = x"
leg_simps_1: shows "C.arr HH\<mu>\<nu>H\<pi>\<rho>.dom.leg0"
normal_divide_x: fixes f :: "real poly" assumes "normal_poly (f * [:0, 1:])" shows "normal_poly f"
nth_imp_genPrefix: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "length xs \<le> length ys"   and "\<forall>i<length xs. (xs ! i, ys ! i) \<in> r" shows "(xs, ys) \<in> genPrefix r"
ideal_radical_ideal: fixes F :: "'c set" shows "ideal (\<surd>ideal F) = \<surd>ideal F"
ipurge_unwinding: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b" assumes "ref_union_closed P" shows "secure P I D = weakly_future_consistent P I D (rel_ipurge P I D)"
ordLeq_refl: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "r \<le>o r"
collapse_1: fixes process :: "'a process" assumes "isAction process" shows "Action (prefOf process) (contOf process) = process"
adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: shows "adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.remove_child get_disconnected_nodes set_disconnected_nodes"
assms: fixes act :: "'act"   and c :: "'a"   and state :: "'state" assumes "bn act \<sharp>* c" shows "P c \<langle>act,state\<rangle>"
mon_fs: shows "\<forall>fi\<in>set fs. monic (Mp fi) \<and> irreducible\<^sub>d_m fi"
cases: fixes x :: "'d"   and \<I> :: "('d, 'e) \<I>"   and a :: "('f, 'd, 'e) gpv" assumes "x \<in> outs_gpv \<I> a" obtains c :: "('f, 'd, 'e) rpv" and gpv :: "('f, 'd, 'e) gpv" where "a = gpva__" and "IO x c \<in> set_spmf (the_gpv gpva__)"   | out :: "'d" and rpv :: "('f, 'd, 'e) rpv" and gpv :: "('f, 'd, 'e) gpv" and input :: "'e" where "a = gpv" and "IO out rpv \<in> set_spmf (the_gpv gpv)" and "input \<in> responses_\<I> \<I> out" and "x \<in> outs_gpv \<I> (rpv input)"
subgroup_mult_equality: fixes H :: "'a set"   and h1 :: "'a"   and h2 :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes>\<^bsub>G\<lparr>carrier := adjoin G H a, carrier := H\<rparr>\<^esub> h2 = h1 \<otimes>\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> h2"
HSem_ignores_fresh_restr: fixes ESem :: "'d \<Rightarrow> ('e \<Rightarrow> 'f) \<rightarrow> 'f"   and S :: "'e set"   and \<Gamma> :: "('e \<times> 'd) list"   and \<rho> :: "'e \<Rightarrow> 'f" assumes "has_ignore_fresh_ESem ESem"   and "atom ` S \<sharp>* \<Gamma>" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<rho> f|` (- S) = has_ESem.HSem ESem \<Gamma>\<cdot>(\<rho> f|` (- S))"
dg_prod_is_arrD_4: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "\<D>\<^sub>\<circ> a = I"
allocationFinite: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G"   and "finite G" shows "finite a"
asAbs_pick: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "goodAbs A" shows "asAbs (pick A) = A"
m_def: shows "0 \<le> m \<and> ord_Zp (f \<ominus> g) = int k + m"
tsubst_preserves_free: fixes \<theta> :: "('d, 'e) trm"   and \<sigma> :: "('d, 'f, 'e) subst" assumes "dfree \<theta>"   and "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "dfree (Tsubst \<theta> \<sigma>)"
that: fixes Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> Q"   and "P' \<sim> Q \<parallel> !P" shows "thesis"
bound: shows "\<forall>\<^sub>F x in at_top. \<forall>y\<in>U. cmod (f y x) \<le> g x"
dfa_reach_rquot: fixes x :: "nat"   and y :: "nat" assumes "dfa_is_node A x"   and "dfa_is_node A y" shows "rquot_memb y (rquot_dfs A n x) = (\<exists>m. dfa_reach A x (zeros m n) y)"
select_convs_1: fixes set_dom :: "'b set"   and set_func :: "'b \<Rightarrow> 'b"   and set_cod :: "'b set"   and more :: "'c" shows "set_dom \<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod, \<dots> = more\<rparr> = set_dom"
inv_solve_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier \<G>"   and "b \<in> carrier \<G>"   and "c \<in> carrier \<G>" shows "(a = inv b \<otimes> c) = (c = b \<otimes> a)"
qe_dlo\<^sub>1_def: fixes as :: "atom list" shows "qe_dlo\<^sub>1 as = (if Less 0 0 \<in> set as then FalseF else let lbs = concat (map (\<lambda>x. case x of Less 0 nat2 \<Rightarrow> [] | Less (Suc i) 0 \<Rightarrow> [i] | Less (Suc i) (Suc nata) \<Rightarrow> [] | Eq nat1 nat2 \<Rightarrow> []) as); ubs = concat (map (\<lambda>x. case x of Less 0 0 \<Rightarrow> [] | Less 0 (Suc j) \<Rightarrow> [j] | Less (Suc nat) nat2 \<Rightarrow> [] | Eq nat1 nat2 \<Rightarrow> []) as) in Let (concat (map (\<lambda>i. map (\<lambda>j. Atom (Less i j)) ubs) lbs)) list_conj)"
isInjAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd" shows "isInjAll h hA \<equiv> isInj h \<and> isInjAbs hA"
reflects_tabulation: fixes r :: "'b"   and f :: "'b"   and \<rho> :: "'b"   and g :: "'b" assumes "D.ide r"   and "D.ide f"   and "\<guillemotleft>\<rho> : g \<Rightarrow>\<^sub>D r \<star>\<^sub>D f\<guillemotright>"   and "tabulation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (F.right_map r) (C.inv (F.right_cmp (r, f)) \<cdot>\<^sub>C F.right_map \<rho>) (F.right_map f) (F.right_map g)" shows "tabulation (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D r \<rho> f g"
less_bit1_def: fixes a :: "'b bit1"   and b :: "'b bit1" shows "(a < b) = (Rep_bit1 a < Rep_bit1 b)"
d_def: shows "d = lcm (fpxs_root_order f) (fpxs_root_order g)"
d_a_export: fixes x :: "'a"   and y :: "'a" shows "am_d (ad x \<cdot> y) = ad x \<cdot> am_d y"
measure_map_spmf: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c spmf"   and A :: "'b set" shows "Sigma_Algebra.measure (measure_spmf (map_spmf f p)) A = Sigma_Algebra.measure (measure_spmf p) (f -` A)"
rep3_def: shows "rep3 \<equiv> prod.fst (prod.snd (prod.snd (fix_syn (\<lambda>(root3, op3, rep3, grep3). (\<Lambda> pat. rep3\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root3\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op3\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. grep3\<cdot>pat\<cdot>(case cfst\<cdot>usvs of [::] \<Rightarrow> Null | u :# us \<Rightarrow> sfoldl\<cdot>(op3\<cdot>pat)\<cdot>(root3\<cdot>pat)\<cdot>us)\<cdot>usvs, \<Lambda> pat l usvs. Node\<cdot>usvs\<cdot>(next\<cdot>(csnd\<cdot>usvs)\<cdot>l)\<cdot>(case csnd\<cdot>usvs of [::] \<Rightarrow> Null | v :# vs \<Rightarrow> rep3\<cdot>pat\<cdot>(cfst\<cdot>usvs :@ [:v:], vs)))))))"
is_processT1: fixes P :: "'c process" shows "([], {}) \<in> F P"
hom_zero: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "ring_hom_cring R S h" shows "h \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>S\<^esub>"
abs_eq: fixes x :: "'b list" shows "fset_from_list x = Abs_fset (set x)"
simps_2: fixes x :: "'b"   and y :: "'b"   and p :: "'c" shows "srcNet (AllowPortFromTo x y p) = x"
assms: shows "(NodeS i p R, a, s') \<in> node_sos S"
set_append: fixes xs :: "'a list"   and ys :: "'a list" shows "list.set (xs @ ys) = list.set xs \<union> list.set ys"
equivclp_into_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "equivclp r a b"   and "r b c \<or> r c b" shows "equivclp r a c"
assms_4: fixes x :: "'s"   and M :: "'s stream measure" assumes "R x M" shows "\<exists>M'. (\<forall>y\<in>set_pmf (K x). R y (M' y)) \<and> (\<forall>y. sets (M' y) = sets (stream_space (count_space UNIV)) \<and> prob_space (M' y)) \<and> M = measure_pmf (K x) \<bind> (\<lambda>y. distr (M' y) (stream_space (count_space UNIV)) ((##) y))"
matrix_equation_main_step: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and I :: "nat list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "list_constr I (length qs)"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "vec_of_list (mtx_row signs I) \<bullet> w_vec p qs signs = construct_NofI p (retrieve_polys qs I)"
n0'_2: shows "\<forall>n\<ge>n\<^sub>0'. s (w \<triangleright> n) \<down>= Suc i'"
sumI: fixes u :: "'b"   and U :: "'b set"   and v :: "'b"   and V :: "'b set"   and x :: "'b" assumes "u \<in> U"   and "v \<in> V"   and "x = u + v" shows "x \<in> U + V"
disc_eq_case_1: fixes option :: "'c option" shows "(option = None) = (case option of None \<Rightarrow> True | Some uu_ \<Rightarrow> False)"
distinct_39: fixes x51 :: "nat"   and x52 :: "nat" shows "disconnect(x51, x52) \<noteq> \<tau>\<^sub>n"
dghm_singleton_components_1: fixes j :: "V"   and \<CC> :: "V" shows "dghm_singleton j \<CC>\<lparr>ObjMap\<rparr> = (\<lambda>a\<in>\<^sub>\<circ>\<CC>\<lparr>Obj\<rparr>. ZFC_in_HOL.set {\<langle>j, a\<rangle>})"
Ag_ind_triangle: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and j :: "'c"   and f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'f"   and B :: "'f set" assumes "\<forall>j\<in>I. aGroup (A j)"   and "j \<in> I"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to f (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "j \<in> I" shows "compos (a\<Pi>\<^bsub>I\<^esub> A) (compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f) (\<pi>\<^bsub>I,A,j\<^esub>) (ainvf\<^bsub>(a\<Pi>\<^bsub>I\<^esub> A),Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f\<^esub> Agii (a\<Pi>\<^bsub>I\<^esub> A) f)) (Agii (a\<Pi>\<^bsub>I\<^esub> A) f) = \<pi>\<^bsub>I,A,j\<^esub>"
not_coprimeE: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "\<not> algebraic_semidom.coprime times one a b" obtains c :: "'a" where "dvd.dvd times c a" and "dvd.dvd times c b" and "\<not> dvd.dvd times c one"
M_bottom: shows "0\<^sub>m 1 n @\<^sub>c - 1\<^sub>m 1 \<in> carrier_mat 1 (n + 1)"
succ_def: fixes r :: "('a \<times> 'a) set"   and i :: "'a" assumes "wo_rel r" shows "succ r i \<equiv> suc r {i}"
hfref_mop_conv: fixes g :: "'e \<Rightarrow> 'f Heap"   and P :: "'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h nres"   and Q :: "'g \<Rightarrow> bool"   and R :: "('g \<Rightarrow> 'e \<Rightarrow> assn) \<times> ('g \<Rightarrow> 'e \<Rightarrow> assn)"   and S :: "'h \<Rightarrow> 'f \<Rightarrow> assn" shows "((g, mop P f) \<in> [Q]\<^sub>a R \<rightarrow> S) = ((g, f) \<in> [\<lambda>x. P x \<and> Q x]\<^sub>a R \<rightarrow> S)"
mempty_right: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and liftemb :: "'e\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'e\<^sub>\<bottom>"   and liftdefl :: "'e itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'e \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'e"   and defl :: "'e itself \<Rightarrow> udom defl"   and mempty :: "'e"   and mappend :: "'e \<rightarrow> 'e \<rightarrow> 'e"   and xs :: "'e" assumes "class.monoid below liftemb liftprj liftdefl emb prj defl mempty mappend" shows "mappend\<cdot>xs\<cdot>mempty = xs"
domA_delete: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list" shows "domA (delete x \<Gamma>) = domA \<Gamma> - {x}"
better_E_TAbs: fixes X :: "tyvrs"   and t12 :: "trm"   and T11 :: "ty"   and T2 :: "ty" shows "(\<lambda>X<:T11. t12) \<cdot>\<^sub>\<tau> T2 \<longmapsto> t12[X \<mapsto>\<^sub>\<tau> T2]"
swap_mu_def: fixes m :: "nat"   and dmu :: "int iarray iarray"   and i :: "nat"   and dmu_i_im1 :: "int"   and dim1 :: "int"   and di :: "int"   and dsi :: "int" shows "swap_mu m dmu i dmu_i_im1 dim1 di dsi = (let im1 = i - 1 in IArray.of_fun (\<lambda>ii. if ii < im1 then dmu !! ii else if i < ii then let dmu_ii = dmu !! ii in IArray.of_fun (\<lambda>j. let dmu_ii_j = dmu_ii !! j in if j = i then (dsi * dmu_ii !! im1 - dmu_i_im1 * dmu_ii_j) div di else if j = im1 then (dmu_i_im1 * dmu_ii_j + dmu_ii !! i * dim1) div di else dmu_ii_j) ii else if ii = i then let mu_im1 = dmu !! im1 in IArray.of_fun (\<lambda>j. if j = im1 then dmu_i_im1 else mu_im1 !! j) ii else IArray.of_fun ((!!) (dmu !! i)) ii) m)"
mult_sheaf_spec_in_sheaf_spec: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set"   and t :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "s \<in> \<O> U"   and "t \<in> \<O> U"   and "U \<subseteq> Spec" shows "mult_sheaf_spec U s t \<in> \<O> U"
vdomain_in_VsetI: fixes r :: "V"   and \<alpha> :: "V" assumes "r \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<D>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"
open_euler1: fixes u :: "'a"   and v :: "'a" assumes "Digraph_Component.connected G"   and "u \<in> verts G"   and "v \<in> verts G"   and "\<And>w. \<lbrakk>w \<in> verts G; u \<noteq> w; v \<noteq> w\<rbrakk> \<Longrightarrow> in_degree G w = out_degree G w"   and "in_degree G u + 1 = out_degree G u"   and "out_degree G v + 1 = in_degree G v" shows "\<exists>p. euler_trail u p v"
sup_least: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "x \<le> z" shows "x \<le> y \<sqinter> z"
tpOf_rOfFax: fixes f :: "'fsym" shows "TE.tpOf (rOfFax f) = resOf f"
restrict_ext: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "restrict f A = restrict g A"
outs_\<I>_uniform: fixes A :: "'d set"   and B :: "'e set" shows "outs_\<I> (\<I>_uniform A B) = (if B = {} then {} else A)"
norms_mus_def: fixes n :: "nat"   and fs :: "'a vec list" shows "gram_schmidt.norms_mus n fs = (let (n_gs, mus) = gram_schmidt.norms_mus' n fs [] [] in (rev n_gs, rev mus))"
3: shows "ss = map ((!) ls) (map (\<lambda>i. SOME j. j < length ls \<and> ls ! j = i) ss)"
prime_normalize_iff: fixes p :: "'a" shows "prime (normalize p) = prime_elem p"
div_neg_pos_less0: fixes a :: "int"   and b :: "int" assumes "a < 0"   and "0 < b" shows "a div b < 0"
m: fixes m :: "int" assumes "poly_mod_type TYPE('b) m" shows "m = int CARD('b)"
simple_ownership_distinct: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list"   and i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'b"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'c"   and \<D>\<^sub>i :: "'d"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "'e"   and p\<^sub>j :: "'b"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'c"   and \<D>\<^sub>j :: "'d"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "'e" assumes "simple_ownership_distinct ts"   and "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "\<O>\<^sub>i \<inter> \<O>\<^sub>j = {}"
PLUS: fixes X :: "freeExp"   and X' :: "freeExp"   and Y :: "freeExp"   and Y' :: "freeExp" assumes "X \<sim> X'"   and "Y \<sim> Y'" shows "PLUS X Y \<sim> PLUS X' Y'"
emeasure_eq_measure: fixes s :: "'s"   and A :: "'s stream set" shows "emeasure (T s) A = ennreal (T.prob s A)"
that: fixes x :: "nat" assumes "x \<in> keys_pp (rep_nat_pp s) \<union> keys_pp (rep_nat_pp t)"   and "comparator_of (lookup_pp (rep_nat_pp s) x) (lookup_pp (rep_nat_pp t) x) = lex_comp' s t"   and "\<And>y. y < x \<Longrightarrow> lookup_pp (rep_nat_pp s) y = lookup_pp (rep_nat_pp t) y" shows "thesis"
map_indets_def: fixes f :: "'d \<Rightarrow> 'f" shows "map_indets f = poly_subst (\<lambda>x. monomial (1::'e) (monomial 1 (f x)))"
restrict_distr: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and \<Omega> :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "\<Omega> \<inter> space N \<in> sets N"   and "f \<in> space M \<rightarrow> \<Omega>" shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"
approx_slp_Elem: fixes e :: "nat \<Rightarrow> real"   and vs :: "real list"   and VS :: "((real \<times> real pdevs) \<times> real) list"   and p :: "nat"   and ra :: "floatarith list"   and X :: "((real \<times> real pdevs) \<times> real) list" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "vs \<in> aforms_err e VS"   and "approx_slp p ra VS = Some X" shows "interpret_slp ra vs \<in> aforms_err e X"
dvd_lcmI2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c" shows "dvd.dvd times a (lcm b c)"