hcju/isabelleps huggingface dataset:
integrable_componentwise: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) integrable_on A" shows "f integrable_on A"
body': shows "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2(pns [\<mapsto>] vs), sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2 ++ l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"
Approximation1_5Tr6: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and a :: "'a"   and x :: "'a"   and j :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "a \<in> carrier K"   and "a \<noteq> \<zero>"   and "x \<in> carrier K"   and "Ostrowski_elem K (Suc n) vv x"   and "j \<in> nset (Suc 0) (Suc n)" shows "\<exists>l. \<forall>m>l. vv j ((1\<^sub>r \<plusminus> -\<^sub>a x)^\<^bsup>K m\<^esup> \<plusminus> a \<cdot>\<^sub>r x^\<^bsup>K m\<^esup>) = 0"
3: shows "\<forall>i k. j \<noteq> k \<longrightarrow> f (i, k) = bot"
word_div_mult: fixes y :: "'b word"   and x :: "'b word" assumes "0 < y"   and "unat x * unat y < 2 ^ LENGTH('b)" shows "x * y div y = x"
STA_le: fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and m :: "nat" assumes "A, n \<turnstile> branch"   and "n \<le> m" shows "A, m \<turnstile> branch"
restrict_map_insert: fixes f :: "'g \<Rightarrow> 'h option"   and a :: "'g"   and A :: "'g set" shows "f |` insert a A = (f |` A)(a := f a)"
comp_simp: fixes f :: "(bool, unit) concrete_category.arr"   and g :: "(bool, unit) concrete_category.arr" assumes "partial_magma.arr (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ()) f g)" shows "concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ()) f g = f"
Err_in_err: fixes A :: "'a set" shows "Err \<in> err A"
th3_2: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2" assumes "B\<^sub>K a b c" shows "B\<^sub>K c b a"
RBT_lookup_empty: fixes t :: "('a, 'c) rbt" shows "(ord.rbt_lookup cless t = Map.empty) = (t = rbt.Empty)"
noreps_def: fixes xs :: "'b list" shows "noreps xs \<equiv> \<forall>i<length xs - 1. xs ! i \<noteq> xs ! Suc i"
typ_substitutions_only_on_free_variables: fixes t :: "typ"   and S :: "nat \<Rightarrow> typ"   and S' :: "nat \<Rightarrow> typ" assumes "\<forall>x\<in>free_tv t. S x = S' x" shows "$ S t = $ S' t"
insert_not_empty: fixes a :: "'g"   and A :: "'g set" shows "insert a A \<noteq> {}"
eq_almost_full_on_finite_set: fixes A :: "'a set" assumes "finite A" shows "almost_full_on (=) A"
l_inv: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "inv x \<otimes> x = \<one>"
diff_fun_submanifold: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'f"   and S :: "'c set" assumes "diff_fun k charts f"   and "open S" shows "diff_fun k (manifold.charts_submanifold charts S) f"
ct_prefix_nil_1: fixes xs :: "ct list" shows "ct_prefix [] xs"
typing_implies_valid: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "llam"   and T :: "ty" assumes " \<Gamma> \<turnstile> t : T " shows "valid \<Gamma>"
expectation_X: fixes i :: "nat" shows "expectation (X i) = p"
monom_mult_assoc: fixes c :: "'b"   and s :: "'x \<Rightarrow>\<^sub>0 nat"   and d :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.monom_mult c s (local.punit.monom_mult d t p) = local.punit.monom_mult (c * d) (s + t) p"
auto_split_is_maxsplit: fixes A :: "'d \<times> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> bool)"   and res :: "'c list \<times> 'c list"   and xs :: "'c list" shows "auto_split A (start A) res [] xs = maxsplit (accepts A) res [] xs"
arr_eqI: fixes f :: "'s"   and f' :: "'s" assumes "par f f'"   and "Fun f = Fun f'" shows "f = f'"
that: assumes "\<sigma>' = \<sigma>"   and "r = Array.get (the_state \<sigma>) a" shows "thesis"
affine_parallel_expl_aux: fixes S :: "'a set"   and a :: "'a"   and T :: "'a set" assumes "\<And>x. (x \<in> S) = (a + x \<in> T)" shows "T = (+) a ` S"
usedBy_lit: fixes x :: "'c \<Longrightarrow> 'd"   and v :: "'e" shows "x \<natural> \<guillemotleft>v\<guillemotright>"
unsigned_of_int: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and k :: "int" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (word_of_int k) = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('c) k))"
isBackjumpLevelEnsuresIsUnitInPrefix: fixes M :: "(Literal \<times> bool) list"   and c :: "Literal list"   and level :: "nat"   and l :: "Literal" assumes "consistent (elements M)"   and "uniq (elements M)"   and "clauseFalse c (elements M)"   and "isBackjumpLevel level l c M" shows "isUnitClause c l (elements (prefixToLevel level M))"
wf_measures: fixes fs :: "('a \<Rightarrow> nat) list" shows "wf (measures fs)"
clI: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> \<E>"   and "\<E>.rank_of (insert x X) = \<E>.rank_of X" shows "x \<in> \<E>.cl X"
measurable_on_componentwise: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "(f measurable_on S) = (\<forall>i\<in>Basis. (\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on S)"
S_def: shows "S \<equiv> segment G.face_cycle_succ a a \<union> segment G.face_cycle_succ a' a'"
valid_as_valid_exec: fixes as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list"   and PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and s :: "('a, bool) fmap" assumes "as \<in> valid_plans PROB"   and "s \<in> valid_states PROB" shows "exec_plan s as \<in> valid_states PROB"
in_hom: shows "\<guillemotleft>\<mu> : r \<Rightarrow> s\<guillemotright>"
elims_3: fixes x :: "\<I>"   and xa :: "nat"   and xb :: "nat \<times> 'a \<times> 'b" assumes "\<not> check_before x xa xb" obtains I :: "\<I>" and dt :: "nat" and t :: "nat" and a :: "'a" and b :: "'b" where "x = I" and "xa = dt" and "xb = (t, a, b)" and "\<not> enat t + right I < enat dt"
one_simplex_segment: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "1 simplex closed_segment a b"
mat_mult_row_code: fixes m :: "(('e, 'f) vec, 'g) vec"   and m' :: "(('e, 'd) vec, 'f) vec"   and f :: "'g" shows "($) (mat_mult_row m m' f) = (\<lambda>c. \<Sum>k\<in>UNIV. m $ f $ k * m' $ k $ c)"
edom_def: fixes m :: "'c \<Rightarrow> 'd" shows "edom m = {x. m x \<noteq> \<bottom>}"
B_def: fixes j :: "nat" shows "B j = {x \<in> S. (real j + 1 / 3) * e \<le> f x}"
nrm_reverse: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "a \<bullet> rev.rot (x - y) = - a \<bullet> rev.rot (y - x)"
left_diff_distrib: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a - b) * c = a * c - b * c"
hyps: fixes v :: "nat" assumes "xs !. v = Some y" shows "(x # xs) !. v = Some y"
gauge_min: fixes E :: "real set"   and g1 :: "real \<Rightarrow> real"   and g2 :: "real \<Rightarrow> real" assumes "gauge E g1"   and "gauge E g2" shows "gauge E (\<lambda>x. min (g1 x) (g2 x))"
star_rtc2: fixes x :: "'a ppath set" shows "(pp_one \<union> pp_prod x x \<subseteq> x) = (x = pp_star x)"
\<zeta>p: shows "\<zeta> \<notin> path_image p"
components_are_iso: fixes a :: "'a" assumes "B.ide a" shows "B.iso (B.\<ll> a)"
c_prod_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<cdot> B = {u \<cdot> v |u v. u \<in> A \<and> v \<in> B}"
fold_mset_add_mset: fixes s :: "nat"   and x :: "'c \<times> 'd"   and M :: "('c \<times> 'd) multiset" shows "fold_mset ((+) \<circ> pair_size) s (add_mset x M) = ((+) \<circ> pair_size) x (fold_mset ((+) \<circ> pair_size) s M)"
cexpr_subst_val_aux_eq_cexpr_subst: fixes x :: "nat"   and e :: "cexpr"   and v :: "val" shows "cexpr_subst_val_aux x e v = cexpr_subst x (CVal v) e"
bind_right_unit: fixes m :: "'a \<Rightarrow> ('b \<times> 'a) option" shows "bind\<^sub>S\<^sub>E m unit\<^sub>S\<^sub>E = m"
learn_finE_2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option" assumes "learn_fin \<psi> U s"   and "f \<in> U" shows "\<exists>i n\<^sub>0. \<psi> i = f \<and> (\<forall>n<n\<^sub>0. s (f \<triangleright> n) \<down>= 0) \<and> (\<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= Suc i)"
Always_def: fixes A :: "'b set" shows "Always A \<equiv> {F. Init F \<subseteq> A} \<inter> Stable A"
Rep_matrix_zero_imp_mult_zero: fixes A :: "'a matrix"   and B :: "'a matrix" assumes "\<forall>j i k. Rep_matrix A j k = (0::'a) \<or> Rep_matrix B k i = (0::'a)" shows "A * B = 0"
fmeasurable_countable_Union: fixes S :: "nat \<Rightarrow> 'a set"   and M :: "'a measure"   and B :: "real" assumes "\<And>i. S i \<in> fmeasurable M"   and "\<And>n. Sigma_Algebra.measure M (\<Union> (S ` {..n})) \<le> B" shows "\<Union> (range S) \<in> fmeasurable M"
thread_oks_ts_change: fixes ts :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and ts' :: "'c \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and tas :: "('c, 'h, 'i) new_thread_action list" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "thread_oks ts tas = thread_oks ts' tas"
\<gamma>: shows "\<gamma> = \<eta> + \<gamma>a"
nzero': fixes i :: "nat" assumes "i < m" shows "\<parallel>fs'.gs.gso i\<parallel>\<^sup>2 \<noteq> 0"
prj_chine_assoc'_1: shows "trg_trg_f.Prj\<^sub>1\<^sub>1 \<cdot> trg_trg_f.chine_assoc' = trg_trg_f.Prj\<^sub>1"
simps_3: fixes uu :: "'a"   and uv :: "bool" shows "restrict_top Falseif uu uv = Falseif"
prems: shows "h \<turnstile> get_dom_component child \<rightarrow>\<^sub>r c"
conSim: shows "weak_reduction_contrasimulation (indRelLTPO TRel) (STCal Source Target)"
isSp_Append_sum: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n''"   and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'" shows "n is-ns @ [n'']\<rightarrow>\<^sub>d* n'"
Polys_closed_minus: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p - q \<in> P[X]"
irreducible_prod_rI: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "Group.monoid G"   and "irreducible G a"   and "b \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "irreducible G (a \<otimes>\<^bsub>G\<^esub> b)"
1: shows "f \<sqinter> - q \<le> - h \<sqinter> - - g"
FI_QUERY_def: fixes P :: "assn"   and Q :: "assn"   and F :: "assn" shows "FI_QUERY P Q F \<equiv> P \<Longrightarrow>\<^sub>A Q * F"
prems_1: fixes u :: "'t" assumes "u \<in> set vs" shows "u \<prec>\<^sub>t v"
add_poly_p_add_to_result: assumes "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#}) ({#}, {#}, r)" shows "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#(ys, n + m)#} + {#}) ({#}, {#}, {#(ys, n + m)#} + r)"
Deterministic_f_Exec_Stream: fixes trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and localState :: "'e \<Rightarrow> 'f"   and c1 :: "'e"   and c2 :: "'e"   and xs :: "'d list" assumes "Deterministic_Trans_Fun trans_fun localState"   and "localState c1 = localState c2" shows "f_Exec_Comp_Stream trans_fun xs c1 = f_Exec_Comp_Stream trans_fun xs c2"
D_def: fixes P :: "'a process" shows "D P = DIVERGENCES (Rep_process P)"
eta_preserves_term_ok: 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 r :: "Core.term"   and s :: "Core.term" assumes "term_ok \<Theta> r"   and "r \<rightarrow>\<^sub>\<eta> s" shows "term_ok \<Theta> s"
abs_mult_sgn: fixes a :: "'a" shows "\<bar>a\<bar> * sgn a = a"
igSubstIPresIGWls_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSubstIPresIGWls MOD \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> igWls MOD s (igSubst MOD ys Y y X)"
triangle_equiv_form_2: shows "((f\<^sup>* \<star> \<epsilon>) \<cdot> \<a>[f\<^sup>*, f, f\<^sup>*] \<cdot> (\<eta> \<star> f\<^sup>*) = \<r>\<^sup>-\<^sup>1[f\<^sup>*] \<cdot> \<l>[f\<^sup>*]) = (\<r>[f\<^sup>*] \<cdot> (f\<^sup>* \<star> \<epsilon>) \<cdot> \<a>[f\<^sup>*, f, f\<^sup>*] \<cdot> (\<eta> \<star> f\<^sup>*) \<cdot> \<l>\<^sup>-\<^sup>1[f\<^sup>*] = f\<^sup>*)"
is_morphism_of_sheaves: shows "morphism_sheaves_of_rings Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf \<phi>\<^sub>f"
landau_function_family_powr_closure: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real" assumes "F \<noteq> bot"   and "filterlim f at_top F" shows "landau_function_family F (powr_closure f)"
elims: fixes x :: "'b action"   and xa :: "'b queue"   and y :: "'b queue" assumes "do_act x xa = y" obtains q :: "'b queue" where "x = Deq" and "xa = qa__" and "y = deq qa__"   | xb :: "'b" and q :: "'b queue" where "x = Enq xb" and "xa = q" and "y = enq xb q"
HoHV_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "H.HoHV \<mu> \<equiv> if VVV.arr \<mu> then (fst \<mu> \<otimes> fst (snd \<mu>)) \<otimes> snd (snd \<mu>) else null"
ode_d1_eq: fixes x :: "'d"   and j :: "'d" shows "ode_d1 x = ode_d 0 x j"
set_integrable_def: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
maximal_def: fixes A :: "'b set"   and P :: "'b set \<Rightarrow> bool" shows "maximal A P = (P A \<and> (\<forall>B. A \<subseteq> B \<and> P B \<longrightarrow> B = A))"
countable_UN: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (\<Union> (A ` I))"
rep_eq: fixes x :: "'b \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c"   and xa :: "('b, 'c) alist"   and xb :: "('b, 'c) alist" shows "alist.impl_of (join x xa xb) = join_raw x (alist.impl_of xa) (alist.impl_of xb)"
var_not_in_linear_fm_helper: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and z :: "nat"   and F :: "atom fm" assumes "var \<notin> vars b"   and "var \<notin> vars c" shows "freeIn (var + z) (linear_substitution_fm_helper var b c F z)"
rreq_oip: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:3..PRreq-:9} \<union> {PRreq-:17, PRreq-:30, PRreq-:32} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>) \<and> (osn \<xi> < sqn (rt \<xi>) (oip \<xi>) \<or> sqn (rt \<xi>) (oip \<xi>) = osn \<xi> \<and> the (dhops (rt \<xi>) (oip \<xi>)) \<le> Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (oip \<xi>)) = val))"
assms_6: shows "has_greatest_postfixpoint (t \<circ> h)"
ownership_distinct_tl: fixes t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "ownership_distinct (t # ts)" shows "ownership_distinct ts"
intro: fixes \<Delta> :: "('a, 'b) network_scheme" assumes "countable \<^bold>E\<^bsub>\<Delta>\<^esub>"   and "source \<Delta> \<noteq> sink \<Delta>"   and "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> capacity \<Delta> e = 0"   and "\<And>e. capacity \<Delta> e \<noteq> \<top>" shows "countable_network \<Delta>"
aes: shows "aes = map (\<lambda>x. (x, 0)) as"
fresh_iff_non_subst_dbtm: fixes i :: "name"   and t :: "dbtm" shows "(subst_dbtm DBZero i t = t) = atom i \<sharp> t"
open_real_greaterThan: fixes a :: "real" shows "open {a<..}"
rvars_of_lvar_rvars: fixes x :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "x \<in> lvars t" shows "rvars_eq (eq_for_lvar t x) \<subseteq> rvars t"
mcb: shows "mat_of_col (- b) \<in> carrier_mat nr 1"
repeatable_many_calls: fixes x :: "'b" shows "repeatable (many_calls x)"
subgraph_threshold_def: fixes H :: "nat set \<times> nat set set"   and n :: "nat" shows "subgraph_threshold H n = real n powr - (1 / max_density H)"
order_unique_lemma: fixes a :: "'b"   and n :: "nat"   and p :: "'b poly" assumes "[:- a, 1::'b:] ^ n dvd p"   and "\<not> [:- a, 1::'b:] ^ Suc n dvd p" shows "order a p = n"
Cauchy_convergent: fixes dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'e" assumes "class.complete_space dist uniformity open"   and "uniform_space.Cauchy uniformity X" shows "topological_space.convergent open X"
add_less_cancel_left: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and c :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus c a) (plus c b) = less a b"
fls_mult_of_int_nth_1: fixes k :: "int"   and f :: "'a fls"   and n :: "int" shows "fls_nth (of_int k * f) n = of_int k * fls_nth f n"
ChamberSubcomplexD_sub: fixes Y :: "'a set set" assumes "folding_g.ChamberSubcomplex Y" shows "order.greater_eq X Y"
inverse_arrows_def: fixes f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" shows "S.inverse_arrows f g \<equiv> S.ide (S.comp g f) \<and> S.ide (S.comp f g)"
\<tau>1: shows "typ_of (Abs_fv x \<tau>' s) = Some (\<tau>' \<rightarrow> \<tau>)"
iterateoi_def: fixes olist_it :: "'c \<Rightarrow> ('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list"   and S :: "'c" shows "poly_set_iterateoi_defs.iterateoi olist_it S \<equiv> it_to_it (olist_it S)"
vrange_vconst_on_vempty: fixes c :: "V" shows "\<R>\<^sub>\<circ> (vconst_on 0 c) = 0"
supp_Pred: fixes \<phi> :: "'b" shows "supp (S_action.Pred \<phi>) = supp \<phi>"
image_cong_simp: fixes M :: "'b set"   and N :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = N"   and "\<And>x. x \<in> N =simp=> f x = g x" shows "f ` M = g ` N"
Range: fixes y :: "ZF"   and f :: "ZF" shows "Elem y (HOLZF.Range f) = (\<exists>x. Elem (Opair x y) f)"
divide_neg_pos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less x zero"   and "less zero y" shows "less (divide x y) zero"
ConsThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and es :: "nat exp list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>throw e' # es,s\<^sub>1\<rangle>"
ncg_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ncg x y"   and "ncg y z" shows "ncg x z"
inverse_negative_iff_negative: fixes a :: "'a" shows "(inverse a < (0::'a)) = (a < (0::'a))"
ln_inverse_approx_le: fixes x :: "real"   and a :: "real" assumes "0 < x"   and "0 < a" shows "ln (x + a) - ln x \<le> a * (inverse x + inverse (x + a)) / 2"
assms_2: fixes b :: "'a" assumes "b \<in> B" shows "\<exists>a\<in>A. b .= a"
simulate_plan'_correct: fixes s :: "nat \<Rightarrow> nat option"   and \<pi>s :: "char list list" assumes "s \<in> valid_states" shows "simulate_plan' problem \<pi>s s = simulate_plan \<pi>s s"
exhaust_sel: fixes tree :: "'a tree" obtains "tree = \<langle>\<rangle>"   | "tree = \<langle>un_Node1 tree, value tree, un_Node3 tree\<rangle>"
loc_sound: shows "\<forall>i<length cms. locally_sound_mode_use (cms ! i, mem)"
vfsequence_vcard_vcons: fixes x :: "V" shows "vcard (xs\<^sub>2 #\<^sub>\<circ> x) = ZFC_in_HOL.succ (vcard xs\<^sub>2)"
scene_union_compl: fixes X :: "'a scene" assumes "idem_scene X" shows "X \<squnion>\<^sub>S - X = \<top>\<^sub>S"
homology_dimension_reduced: fixes X :: "'a topology"   and a :: "'a"   and p :: "int" assumes "topspace X = {a}" shows "trivial_group (reduced_homology_group p X)"
assms: shows "\<forall>x\<in>set xs. (Q\<cdot>x andalso P\<cdot>x) = (P\<cdot>x andalso Q\<cdot>x)"
valid_node_def: fixes n :: "'b LDCFG_node" shows "CFGExit.valid_node n \<equiv> \<exists>a. lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a \<and> (n = src a \<or> n = trg a)"
assms_2: fixes i :: "nat" shows "(0::'a) \<le> coeff p i"
D6: fixes r :: "j \<Rightarrow> i \<Rightarrow> bool"   and \<phi> :: "\<o>"   and w :: "i" shows "Some r = d\<^sub>0 (\<^bold>\<lambda>\<^sup>0 \<phi>) \<longrightarrow> ex0 r w = [\<phi> in w]"
ennreal_SUP_add_right: fixes I :: "'a set"   and c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "I \<noteq> {}" shows "c + Sup (f ` I) = (SUP i\<in>I. c + f i)"
n: shows "\<langle>M; n\<rangle> \<turnstile>\<^sub>n Fun f T"
z_neq_inf: fixes z :: "int" shows "ant z \<noteq> \<infinity>"
declclass_methdMembr_simp: fixes m :: "qtname \<times> mdecl" shows "declclass (methdMembr m) = declclass m"
weakly_preserves_objects: fixes a :: "'a" assumes "C.obj a" shows "D.isomorphic (F.map\<^sub>0 a) (F a)"
ToCT_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "T.ToCT f \<equiv> if CCC.arr f then T (fst f, T (fst (snd f), snd (snd f))) else null"
sub_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<squnion> (- y \<squnion> - z) = - x \<squnion> - y \<squnion> - z"
a: shows "rep_list (monomial (1::'b) (term_of_pair (0::'a, i))) = fs ! i"
reflp_def: fixes r :: "'x \<Rightarrow> 'x \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
closest_def: fixes v :: "'a \<Rightarrow> 'b"   and p :: "'a \<Rightarrow> bool"   and s :: "'a set"   and x :: "'b" shows "closest v p s x = (SOME a. is_closest v s x a \<and> ((\<exists>b. is_closest v s x b \<and> p b) \<longrightarrow> p a))"
InvariantVarsQ_def: fixes Q :: "Literal list"   and F0 :: "Literal list list"   and Vbl :: "nat set" shows "InvariantVarsQ Q F0 Vbl \<equiv> vars Q \<subseteq> vars F0 \<union> Vbl"
order_less_le_subst2: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "a < b"   and "f b \<le> c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "f a < c"
qtable_wf_tupleD: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" assumes "qtable n A P Q X" shows "\<forall>x\<in>X. wf_tuple n A x"
dims_tensor_from_lookup: fixes ds :: "nat list"   and e :: "nat list \<Rightarrow> 'a" shows "dims (tensor_from_lookup ds e) = ds"
Abs_finfun_cases: fixes x :: "('a, 'b) finfun" obtains y :: "'a \<Rightarrow> 'b" where "x = Abs_finfun y" and "y \<in> finfun"