hcju/isabelleps huggingface dataset:
continuous_map_subtopology_eu: fixes S :: "'b set"   and T :: "'c set"   and h :: "'b \<Rightarrow> 'c" shows "continuous_map (top_of_set S) (top_of_set T) h = (continuous_on S h \<and> h ` S \<subseteq> T)"
measurable_bounded_by_integrable_imp_integrable_real: fixes f :: "'c \<Rightarrow> real"   and S :: "'c set"   and g :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<le> g x"   and "S \<in> sets lebesgue" shows "f integrable_on S"
qe_FMo\<^sub>1_def: fixes as :: "atom list" shows "qe_FMo\<^sub>1 as = list_conj (concat (map (\<lambda>p. map (\<lambda>q. asimp (combine p q)) (ubounds as)) (lbounds as)))"
\<theta>_3: shows "subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<theta>) \<inter> (subterms s \<union> subterms t) = {}"
stake_nth: fixes m :: "nat"   and n :: "nat"   and s :: "'b stream" assumes "m < n" shows "stake n s ! m = s !! m"
surjective_pairing: fixes t :: "'e \<times> 'f" shows "t = (fst t, snd t)"
cr_vrat_def: fixes a :: "V"   and b :: "rat" shows "cr_vrat a b = (a = b\<^sub>\<rat>)"
has_derivative_locally_injective: fixes a :: "'b"   and S :: "'b set"   and g' :: "'c \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "a \<in> S"   and "open S"   and "bounded_linear g'"   and "g' \<circ> f' a = id"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x)"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e" obtains r :: "real" where "0 < r" and "ball a r \<subseteq> S" and "inj_on f (ball a r)"
comp_assoc: fixes h :: "('o, 'a) arr"   and g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" shows "COMP (COMP h g) f = COMP h (COMP g f)"
exhaust_sel: fixes list :: "'c list" obtains "list = \<epsilon>"   | "list = hd list # tl list"
when_cong: fixes P :: "bool"   and Q :: "bool"   and a :: "'a"   and b :: "'a" assumes "P = Q"   and "Q \<Longrightarrow> a = b" shows "(a when P) = (b when Q)"
zcount_eq_zero_iff: fixes M :: "'c zmultiset"   and x :: "'c" shows "(zcount M x = 0) = (x \<notin>#\<^sub>z M)"
n_rw: shows "nrows_iarray (IArray (map (\<lambda>x. IArray (map (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) [0..<CARD('b)])) [0..<CARD('b)])) = CARD('b)"
functor_implies_equals_cod: assumes "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) A.map" shows "A.map = A.map"
update_convs_2: fixes pc_oiface' :: "char list \<Rightarrow> char list"   and pc_iiface :: "char list"   and pc_oiface :: "char list"   and pc_proto :: "8 word"   and pc_sport :: "16 word"   and pc_dport :: "16 word"   and more :: "'a" shows "pc_oiface_update pc_oiface' \<lparr>pc_iiface = pc_iiface, pc_oiface = pc_oiface, pc_proto = pc_proto, pc_sport = pc_sport, pc_dport = pc_dport, \<dots> = more\<rparr> = \<lparr>pc_iiface = pc_iiface, pc_oiface = pc_oiface' pc_oiface, pc_proto = pc_proto, pc_sport = pc_sport, pc_dport = pc_dport, \<dots> = more\<rparr>"
scaleR: fixes f :: "'a \<Rightarrow> 'b"   and r :: "real"   and b :: "'a" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
elims_2: fixes x :: "('b ivl \<times> 'b) tree" assumes "invar x" obtains t :: "('b ivl \<times> 'b) tree" where "x = t" and "inv_max_hi t \<and> Sorted_Less.sorted (Tree2.inorder t)"
add_mset_remove_trivial_If: fixes a :: "'b"   and N :: "'b multiset" shows "a, N - (a, {#}) = (if a \<in># N then N else a, N)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "idom_hom hom"   and "zero_hom_0 hom" shows "inj_idom_hom hom"
arccos_cos_eq_abs_2pi: fixes \<theta> :: "real" obtains k :: "int" where "arccos (cos \<theta>) = \<bar>\<theta> - real_of_int k * (2 * pi)\<bar>"
mult_assoc: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" shows "(a \<cdot> b) \<cdot> c = a \<cdot> b \<cdot> c"
liear_comb_memTr: fixes A :: "'b set"   and H :: "'a set"   and n :: "nat" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "\<forall>s m. s \<in> {j. j \<le> n} \<rightarrow> A \<and> m \<in> {j. j \<le> n} \<rightarrow> H \<longrightarrow> l_comb R M n s m \<in> carrier M"
lower_asymptotic_density_in_01_2: fixes A :: "nat set" shows "lower_asymptotic_density A \<le> 1"
polynomial_of_mset_mult_map: fixes xs :: "char list multiset"   and m :: "int"   and q :: "(char list multiset \<times> int) multiset" shows "polynomial_of_mset {#case x of (ys, n) \<Rightarrow> (remdups_mset (ys + xs), n * m). x \<in># q#} - Const m * (poly_of_vars xs * polynomial_of_mset q) \<in> More_Modules.ideal polynomial_bool"
rotate_drop_take: fixes n :: "nat"   and xs :: "'a list" shows "rotate n xs = drop (n mod |xs|) xs @ take (n mod |xs|) xs"
prefixeq_butlast: fixes xs :: "'e list" shows "prefix (butlast xs) xs"
carrier_direct_lim_def: shows "stk2.carrier_direct_lim \<equiv> e2.Partition"
const_type_inv_wf: fixes c :: "'fun"   and X :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and a :: "'atom" assumes "\<Gamma> (Fun c X) = Var a"   and "wf\<^sub>t\<^sub>r\<^sub>m (Fun c X)" shows "X = []"
entt_def_true: fixes P :: "assn"   and Q :: "assn" shows "P \<Longrightarrow>\<^sub>t Q \<equiv> P * true \<Longrightarrow>\<^sub>A Q * true"
qGoodInp_def: fixes qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option" shows "qGoodInp qinp \<equiv> liftAll qGood qinp \<and> |{i. qinp i \<noteq> None}| <o |UNIV|"
enumerate_eq_zip: fixes n :: "nat"   and xs :: "'f list" shows "enumerate n xs = zip [n..<n + length xs] xs"
crel_vs_executeD: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "['c]"   and heap :: "heap" assumes "crel_vs R a b"   and "P heap"   and "Q heap"   and "state_dp_consistency.cmem heap" obtains x :: "'c" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "Q heap'" and "state_dp_consistency.cmem heap'" and "R a x"
if_P: fixes P :: "bool"   and x :: "'d"   and y :: "'d" assumes "P" shows "(if P then x else y) = x"
usubstappf_det: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<phi> :: "fml"   and V :: "variable set" assumes "usubstappf \<sigma> U \<phi> \<noteq> undeff"   and "usubstappf \<sigma> V \<phi> \<noteq> undeff" shows "usubstappf \<sigma> U \<phi> = usubstappf \<sigma> V \<phi>"
\<alpha>: shows "\<^bold>g [^] \<alpha> = h0"
u: fixes i :: "nat" shows "u i \<in> borel_measurable M"
simstep_disj1: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "s \<approx> t" shows "\<exists>m. s |\<^sub>s n \<approx> t |\<^sub>s m"
p: fixes C :: "char list" assumes "\<forall>D. (C, D) \<in> (subcls1 P)\<^sup>+ \<longrightarrow> Q D" shows "Q C"
assms_3: fixes c :: "real"   and x :: "'a" assumes "c \<noteq> 0"   and "x \<in> F" shows "c *\<^sub>R x \<in> F"
scons_eq_shift: fixes a :: "'b"   and w :: "'b stream"   and u :: "'b list"   and v :: "'b stream" shows "(a ## w = u @- v) = ([] = u \<and> a ## w = v \<or> (\<exists>u'. a # u' = u \<and> w = u' @- v))"
measurable_shd: fixes M :: "'a measure" shows "shd \<in> stream_space M \<rightarrow>\<^sub>M M"
fold_insert: fixes A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
inv_into_f_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and y :: "'c" assumes "inj_on f A"   and "x \<in> A"   and "f x = y" shows "inv_into A f y = x"
one_Integer_neq_bottom: shows "1 \<noteq> \<bottom>"
lift_def: fixes f :: "'e \<Rightarrow> 'a"   and w :: "(bool \<times> 'e) list" shows "G.lift f w = G.m_concat (map (G.lift_gi f) w)"
star_right_induct: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<squnion> x \<sqinter> y \<le> x \<longrightarrow> z \<sqinter> top \<le> x"
coprime_power_left_iff: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one (power.power one times a n) b = (algebraic_semidom.coprime times one a b \<or> n = 0)"
union_inter_neutral: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "\<forall>x\<in>A \<inter> B. g x = \<^bold>1" shows "F g (A \<union> B) = F g A \<^bold>* F g B"
eq_E_simple_2: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" shows "[x \<^bold>=\<^sub>E y \<^bold>\<rightarrow> x \<^bold>= y in v]"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i < length xs" shows "xs[i := x] ! i = x"
sets_1: fixes i :: "'a" assumes "i \<in> I" shows "sets (M i) = sets N"
nb\<^sub>1: fixes op :: "('a, 'b) sas_plus_operator"   and sa :: "'a \<Rightarrow> 'b option" assumes "op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)" shows "\<phi>\<^sub>S \<Psi> sa \<then> \<phi>\<^sub>O \<Psi> op = \<phi>\<^sub>S \<Psi> sa \<then>\<^sub>+ op"
min_diff_distrib_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "min x y - z = min (x - z) (y - z)"
ax: shows "ax = \<lbrakk>quot_dbfm (abst_dbfm (decode_Var v) 0 B)\<rbrakk>e"
fmeasurable_eq_sets: shows "fmeasurable P = local.P.events"
Action: fixes \<alpha> :: "act"   and P :: "ccs" shows "\<alpha>.P \<longmapsto>\<alpha> \<prec> P"
simps: fixes x :: "tm" shows "Equality_axP x = x EQ \<guillemotleft>refl_ax\<guillemotright> OR x EQ \<guillemotleft>eq_cong_ax\<guillemotright> OR x EQ \<guillemotleft>mem_cong_ax\<guillemotright> OR x EQ \<guillemotleft>eats_cong_ax\<guillemotright>"
is_sublist_at4: fixes as :: "'a list"   and bs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct (as @ bs)"   and "is_sublist [x, y] (as @ bs)"   and "as \<noteq> []"   and "x = last as" shows "y = hd bs"
reflexive: fixes P :: "pi" shows "P \<sim>\<^sub>e P"
uminus_zero_vec: fixes n :: "nat" shows "- 0\<^sub>v n = 0\<^sub>v n"
assms_2: shows "y \<cdot> p\<^sup>o \<le> x"
of_weak_ranking_weak_ranking_aux': fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "finite_total_preorder_on carrier le"   and "A \<subseteq> carrier"   and "x \<in> A"   and "y \<in> A" shows "of_weak_ranking (finite_total_preorder_on.weak_ranking_aux carrier le A) x y = restrict_relation A le x y"
rel_Grp: fixes A1 :: "'h set"   and f1 :: "'h \<Rightarrow> 'j"   and A2 :: "'i set"   and f2 :: "'i \<Rightarrow> 'k" shows "rel_prod (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. Basic_BNFs.fsts x \<subseteq> A1 \<and> Basic_BNFs.snds x \<subseteq> A2} (map_prod f1 f2)"
treap_rotate2: fixes l :: "('a \<times> 'b) tree"   and l2 :: "('a \<times> 'b) tree"   and r2 :: "('a \<times> 'b) tree"   and p1 :: "'b"   and p2 :: "'b"   and k1 :: "'a"   and k :: "'a"   and p :: "'b"   and r :: "('a \<times> 'b) tree"   and k2 :: "'a" assumes "treap l"   and "treap l2"   and "treap r2"   and "\<not> p1 \<le> p2"   and "k1 < k"   and "ins k p r = \<langle>l2, (k2, p2), r2\<rangle>"   and "treap (ins k p r)"   and "treap \<langle>l, (k1, p1), r\<rangle>" shows "treap \<langle>\<langle>l, (k1, p1), l2\<rangle>, (k2, p2), r2\<rangle>"
map_mem_clw_rel_br: fixes a :: "'e \<Rightarrow> 'c set"   and f :: "'d \<Rightarrow> 'e"   and xs :: "'d list"   and X :: "'c set"   and I :: "'e \<Rightarrow> bool" assumes "(\<Union>x\<in>set xs. a (f x)) = X"   and "\<And>x. x \<in> set xs \<Longrightarrow> I (f x)" shows "(map f xs, X) \<in> clw_rel (br a I)"
simps_2: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a" shows "iterate (Suc n) f = f \<circ> iterate n f"
1: shows "one_final_but g (if r = u then {} else Edges (r # between (vertices f) r u @ [u]))"
hyps_8: shows "fst (gb_schema_dummy (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data')) = gb_schema_aux (rc, data') (ab gs bs hs data') (ap gs bs (ps -- sps) hs data')"
nonEqualitySetOfSets: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<noteq> Y" shows "{{x} |x. x \<in> X} \<noteq> {{x} |x. x \<in> Y}"
pre_subdivFace'_distinct: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and v :: "nat"   and n :: "nat"   and vol :: "nat option list" assumes "pre_subdivFace' g f v' v n vol" shows "distinct (removeNones vol)"
hom_add: fixes x :: "'c"   and y :: "'c" shows "g (x + y) = g x + g y"
empty_iff: fixes c :: "'b" shows "(c \<in> {}) = False"
filter_eq_replicate_mset: fixes x :: "'c"   and D :: "'c multiset" shows "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
compile_env_entry_def: fixes Finca_add :: "'l \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef \<Rightarrow> 'l"   and Finca_to_list :: "'l \<Rightarrow> ('m \<times> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef) list"   and Fubx_empty :: "'r"   and Fubx_get :: "'r \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef option"   and Fubx_add :: "'r \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef \<Rightarrow> 'r"   and Fubx_to_list :: "'r \<Rightarrow> ('m \<times> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef) list"   and heap_empty :: "'v"   and heap_get :: "'v \<Rightarrow> 'o \<times> 'n \<Rightarrow> 'n option"   and heap_add :: "'v \<Rightarrow> 'o \<times> 'n \<Rightarrow> 'n \<Rightarrow> 'v"   and heap_to_list :: "'v \<Rightarrow> (('o \<times> 'n) \<times> 'n) list"   and is_true :: "'n \<Rightarrow> bool"   and is_false :: "'n \<Rightarrow> bool"   and box_ubx1 :: "'t \<Rightarrow> 'n"   and unbox_ubx1 :: "'n \<Rightarrow> 't option"   and box_ubx2 :: "'u \<Rightarrow> 'n"   and unbox_ubx2 :: "'n \<Rightarrow> 'u option"   and \<OO>\<pp> :: "'p \<Rightarrow> 'n list \<Rightarrow> 'n"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'p \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'q \<Rightarrow> 'n list \<Rightarrow> 'n"   and \<II>\<nn>\<ll> :: "'p \<Rightarrow> 'n list \<Rightarrow> 'q option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'q \<Rightarrow> 'n list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'q \<Rightarrow> 'p"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'s \<Rightarrow> ('n, 't, 'u) unboxed list \<Rightarrow> ('n, 't, 'u) unboxed option"   and \<UU>\<bb>\<xx> :: "'q \<Rightarrow> Unboxed.type option list \<Rightarrow> 's option"   and \<BB>\<oo>\<xx> :: "'s \<Rightarrow> 'q"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'s \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and Finca_empty :: "'l"   and Finca_get :: "'l \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef option"   and \<A> :: "'m \<Rightarrow> nat option"   and \<O> :: "'w \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and x :: "'w \<times> ('n, 'x, 'm, 'p, 'q) Inca.instr fundef" assumes "inca_ubx_compiler Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> Finca_empty Finca_get" shows "inca_ubx_compiler.compile_env_entry unbox_ubx1 unbox_ubx2 \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> \<A> \<O> x = map_result id (Pair (fst x)) (inca_to_ubx_simulation.compile_fundef unbox_ubx1 unbox_ubx2 \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> \<A> (\<O> (fst x)) (snd x))"
absorb_iff2: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry" shows "(b \<preceq> a) = (linordered_monoid.min a b = b)"
all_eqI: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "(\<forall>x. P x) = (\<forall>x. Q x)"
pred_sexp_trans2: fixes x :: "('b, unit) Old_Datatype.node set"   and y :: "('b, unit) Old_Datatype.node set"   and M :: "('b, unit) Old_Datatype.node set" assumes "(x, y) \<in> pred_sexp\<^sup>+"   and "M \<in> sexp"   and "y \<in> sexp" shows "(x, Scons M y) \<in> pred_sexp\<^sup>+"
nAct: shows "\<not> (\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)"
reduced_word_problem: fixes w :: "'w"   and ss :: "'w list"   and ts :: "'w list" assumes "w \<noteq> (0::'w)"   and "S_reduced_for w ss"   and "S_reduced_for w ts" shows "\<exists>xss. flip_altsublist_chain (ss # xss @ [ts])"
del_left_mset: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "del_left t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "x \<in># mset_tree t \<and> mset_tree t' = mset_tree t - {#x#}"
fixing_arr_gives_natural_transformation_1: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd"   and f1 :: "'a" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A1 f1" shows "natural_transformation A2 B (\<lambda>f2. F (partial_magma.dom A1 f1, f2)) (\<lambda>f2. F (partial_magma.cod A1 f1, f2)) (\<lambda>f2. F (f1, f2))"
simps_3: fixes xs :: "'b list"   and uv :: "'b list" shows "newton_coefficients_main xs [] uv = []"
strongly_strategyproof_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "strongly_strategyproof_sds agents alts sds \<equiv> social_decision_scheme agents alts sds \<and> strongly_strategyproof_sds_axioms agents alts sds"
infinity: fixes undiago :: "'a \<Rightarrow> nat"   and diago :: "nat \<Rightarrow> 'a"   and S :: "'a set" assumes "\<forall>n. undiago (diago n) = n"   and "\<forall>n. diago n \<in> S" shows "infinite S"
simplicial_simplex_id: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set" shows "simplicial_simplex p S (restrict id (standard_simplex p)) = (standard_simplex p \<subseteq> S)"
map\<^sub>0_simps_1: fixes a :: "'c" assumes "C.obj a" shows "D.obj (G.map\<^sub>0 a)"
g': shows "g' \<in> UNIV \<rightarrow> {- 1..1}"
subspace_direct_decomp: fixes A :: "('a \<Rightarrow>\<^sub>0 'b) set"   and ss :: "('a \<Rightarrow>\<^sub>0 'b) set list" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> phull.subspace s" shows "phull.subspace A"
RPS_def: shows "RPS = {p \<in> set PS. (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<in> R}"
per_init_def: fixes D :: "'a set" shows "per_init D \<equiv> {(i, i) |i. i \<in> D}"
scalar_product_def: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec"   and w :: "('d, 'e) Finite_Cartesian_Product.vec" shows "scalar_product v w = (\<Sum>i\<in>UNIV. v $h i * w $h i)"
integrable_I: fixes n :: "nat" shows "(\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}"
assms_3: shows "y \<squnion> y\<^sup>T = x"
bot: fixes x :: "'b"   and xa :: "'c" assumes "(x, xa) \<in> Rb" shows "(lub {}, fixp_fun F' xa) \<in> Ra"
step'_Com: fixes S :: "ivl Abs_State.st option"   and X :: "char list set"   and c :: "ivl Abs_State.st option acom" assumes "domo S \<subseteq> X"   and "vars (strip c) \<subseteq> X"   and "c \<in> Com X" shows "step_ivl S c \<in> Com X"
subst_bvs1'_App: fixes s :: "Core.term"   and t :: "Core.term"   and k :: "nat"   and args :: "Core.term list" shows "subst_bvs1' (s $ t) k args = subst_bvs1' s k args $ subst_bvs1' t k args"
det_iarrays_rings_def: fixes A :: "'b iarray iarray" shows "det_iarrays_rings A = (let A' = echelon_form_of_det_iarrays A euclid_ext2 in (1::'b) div fst A' * (\<Prod>i\<leftarrow>[0..<nrows_iarray A]. snd A' !! i !! i))"
oracle_flag_def: shows "oracle_flag = (\<lambda>key (flg, L) (y, t). return_spmf ((), flg \<or> t = hash key y \<and> y \<notin> L, L))"
dist_vec_nth_le: fixes x :: "('b, 'c) vec"   and i :: "'c"   and y :: "('b, 'c) vec" shows "dist (x $ i) (y $ i) \<le> dist x y"
prv_prv_eql_sym: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and t1 :: "'b"   and t2 :: "'b" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "prv (eql t1 t2)" shows "prv (eql t2 t1)"
cblinfun_eq_on_UNIV_span: fixes basis :: "'a set"   and F :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and G :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "cspan basis = UNIV"   and "\<And>s. s \<in> basis \<Longrightarrow> F *\<^sub>V s = G *\<^sub>V s" shows "F = G"
assms: shows "incseq (\<lambda>n y. f n (x, y))"
rs: shows "\<bar>r - s\<bar> < 2 * pi"
mk_minsky_wit_def: fixes M :: "(nat \<times> (nat, nat) Op) set"   and P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" shows "mk_minsky_wit M P \<equiv> finite M \<and> deterministic M \<and> 0 \<notin> fst ` M \<and> (\<forall>vs. \<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>* \<and> P vs vs')"
ideal_prod_prime: fixes I :: "'a set"   and J :: "'a set"   and P :: "'a set" assumes "ideal R I"   and "ideal R J"   and "prime_ideal R P"   and "I \<diamondsuit>\<^sub>r J \<subseteq> P" shows "I \<subseteq> P \<or> J \<subseteq> P"
intros: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and u :: "'a list"   and v :: "'a list" assumes "traces ind"   and "ind a b" shows "traces.eq_swap ind (u @ [a] @ [b] @ v) (u @ [b] @ [a] @ v)"
minus_apply: fixes A :: "'c \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(A - B) x = A x - B x"
higher_differentiable_on_compose': fixes T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "higher_differentiable_on T f n"   and "higher_differentiable_on S g n"   and "g ` S \<subseteq> T"   and "open S"   and "open T" shows "higher_differentiable_on S (\<lambda>x. f (g x)) n"
ideal_of_variety_of_ideal: fixes A :: "('c \<Rightarrow> 'd) set" shows "\<I> (\<V> (\<I> A)) = \<I> A"
eq_refl_Integer: fixes x :: "Integer" assumes "x \<noteq> \<bottom>" shows "eq\<cdot>x\<cdot>x = TT"
ccApprox_belowI: fixes t :: "Vars.var ttree"   and G :: "CoCalls" assumes "\<And>xs. xs \<in> paths t \<Longrightarrow> ccFromList xs \<sqsubseteq> G" shows "ccApprox t \<sqsubseteq> G"
norm: fixes k :: "nat" shows "cmod (\<Phi> k) = (if coprime k c then 1 else 0)"
bisim_def: shows "bisim \<equiv> {(xa, x). bisimp xa x}"
wf_levellist_def: fixes t :: "dag"   and levellist_old :: "ref list list"   and levellist_new :: "ref list list"   and var :: "ref \<Rightarrow> nat" shows "wf_levellist t levellist_old levellist_new var = (case t of Tip \<Rightarrow> levellist_old = levellist_new | Node lt p rt \<Rightarrow> (\<forall>q. q \<in> set_of t \<longrightarrow> q \<in> set (levellist_new ! var q)) \<and> (\<forall>i\<le>var p. \<exists>prx. levellist_new ! i = prx @ levellist_old ! i \<and> (\<forall>pt\<in>set prx. pt \<in> set_of t \<and> var pt = i)) \<and> (\<forall>i>var p. levellist_new ! i = levellist_old ! i) \<and> length levellist_new = length levellist_old)"