hcju/isabelleps huggingface dataset:
to_nat_less_card: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and j :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep j < CARD('b)"
map_le_upds: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option"   and as :: "'a list"   and bs :: "'b list" assumes "f \<subseteq>\<^sub>m g" shows "f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)"
cost': fixes ts :: "'t list"   and f :: "'op" assumes "\<forall>s\<in>set ts. inv' s"   and "length ts = arity f" shows "cost' f ts = cost f (map hom ts)"
validt_augment_Faults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and F' :: "'f set" assumes "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "F \<subseteq> F'" shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F'\<^esub> P c Q,A"
matchers_map_not_empty_2: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option" shows "{} \<noteq> matchers_map \<sigma>"
assms: fixes x :: "'a" assumes "x \<in># A" shows "f x = g x"
l7_3: fixes M :: "'p"   and A :: "'p" assumes "M Midpoint A A" shows "M = A"
1: shows "f = f \<ominus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f) \<oplus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f)"
Fun_\<Phi>: fixes f :: "'t"   and x :: "'t" assumes "S'.arr f"   and "x \<in> S'.set (S'.dom f)" shows "S.Fun (\<Psi> f) (\<psi> x) = \<Psi>a f (\<psi> x)"
infsetsum_Diff: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and A :: "'b set" assumes "f abs_summable_on B"   and "A \<subseteq> B" shows "infsetsum f (B - A) = infsetsum f B - infsetsum f A"
polypow_normh: fixes p :: "poly"   and n :: "nat"   and k :: "nat" assumes "isnpolyh p n" shows "isnpolyh (p ^\<^sub>p k) n"
slide_def: fixes x :: "wall"   and y :: "wall" shows "slide x y \<equiv> \<exists>B. x = basic (make_vert_block (nat (domain_block B))) \<circ> basic B \<and> y = basic B \<circ> basic (make_vert_block (nat (codomain_block B))) \<and> domain_block B \<noteq> 0"
infty_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>infty_rel = {(Num a, Num a') |a a'. (a, a') \<in> R} \<union> {(infty.Infty, infty.Infty)}"
subst_neg: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "subst (neg \<phi>) t x = neg (subst \<phi> t x)"
routing_port_ranges_sound: fixes e :: "char list \<times> 'b wordinterval"   and tbl :: "'b routing_rule list"   and s :: "'b wordinterval"   and k :: "'b word" assumes "e \<in> set (routing_port_ranges tbl s)"   and "k \<in> wordinterval_to_set (snd e)"   and "valid_prefixes tbl" shows "fst e = output_iface (routing_table_semantics tbl k)"
span_subset_spanI: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> ideal B" shows "ideal A \<subseteq> ideal B"
that_1: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> P[X]"   and "g \<noteq> 0"   and "lcf g = (1::'a)"   and "punit.reduced_GB F = {g}" shows "thesis"
Sup_exp_upper: fixes P :: "'a \<Rightarrow> real"   and S :: "('a \<Rightarrow> real) set"   and b :: "real" assumes "P \<in> S"   and "\<forall>P\<in>S. bounded_by b P" shows "P \<tturnstile> Sup_exp S"
assms: shows "Q \<in> coarser_partitions_with elem P"
r_null: fixes x :: "'c" assumes "x \<in> carrier S" shows "x \<otimes>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub> = \<zero>\<^bsub>S\<^esub>"
intros_2: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and f :: "'f" assumes "s \<in> g"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow> t"
assms: shows "wf_effect_inst (ground_action.effect g_opr)"
assms_2: shows "\<forall>i. f i \<le> a"
dbi_dim: shows "db + i < dim_vec (b @\<^sub>v c)"
image_uminus_greaterThanLessThan: fixes 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 x :: "'a"   and y :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.greaterThanLessThan less x y = ord.greaterThanLessThan less (uminus y) (uminus x)"
set_of_def: fixes S :: "'b" shows "set_of S \<equiv> {x. isNode x \<and> memb x S}"
sound_mode_use\<^sub>A: shows "abs.sound_mode_use (cms\<^sub>A, mem\<^sub>A)"
rep_eq: fixes x :: "'b iso"   and xa :: "'b iso" shows "Rep_iso (x \<cdot> xa) = Rep_iso x \<circ> Rep_iso xa"
dominatesHD: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow>\<^bsub>H\<^esub> y" shows "proj_verts_H x \<rightarrow>\<^sup>*\<^bsub>G\<^esub> proj_verts_H y"
openin_INT: fixes I :: "'c set"   and T :: "'d topology"   and U :: "'c \<Rightarrow> 'd set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)" shows "openin T (\<Inter> (U ` I) \<inter> topspace T)"
assms: fixes s :: "'a" assumes "s \<in> reachable A I" shows "P s"
integrable_Im: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> complex" assumes "complex_integrable M f" shows "integrable M (\<lambda>x. Im (f x))"
power_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and i :: "nat" assumes "z \<squnion> x \<cdot> y \<le> y" shows "x ^ i \<cdot> z \<le> y"
weak_reduction_simulation_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_simulation Rel Cal" shows "weak_reduction_simulation (Rel\<^sup>*) Cal"
simps_3: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c dres" shows "dRETURN x \<bind> f = f x"
no_cycles_implies_unique_paths: fixes v :: "'a"   and w :: "'a" assumes "\<And>xs. \<not> cycle xs"   and "v \<rightarrow>\<^sup>* w" shows "\<exists>!xs. v \<leadsto>xs\<leadsto> w"
swapVal_def: fixes zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b" shows "val ^[z1 \<and> z2]_zs \<equiv> \<lambda>xs x. val xs (x @xs[z1 \<and> z2]_zs)"
ts_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet A B P Q" shows "A \<noteq> B \<and> A \<noteq> P \<and> A \<noteq> Q \<and> B \<noteq> P \<and> B \<noteq> Q \<and> P \<noteq> Q"
RedInitBlock: fixes 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 T :: "ty"   and v :: "val"   and v' :: "val"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and u :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> T casts v to v' " shows "P,E \<turnstile> \<langle>{V:T; V:=Val v;; Val u},s\<rangle> \<rightarrow> \<langle>Val u,s\<rangle>"
F: shows "(\<Sum>a\<in>(\<Union>x. set_pmf (f x)). a * pmf (f False) a) = E (f False)"
monom_eq_var_pow: fixes K :: "'a list set"   and a :: "'a list"   and n :: "nat" assumes "subring K (K [X])"   and "a \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}" shows "UP.monom a n = [a] \<otimes>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> X\<^bsub>K [X]\<^esub> [^]\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> n"
simps_1: fixes x :: "Preliminaries.var" shows "Ik.tpOf (Var x) = GE.tpOfV x"
inf_sup_distrib2: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and z :: "'c"   and x :: "'c" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
cond_case_prod_eta: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i"   and g :: "'g \<times> 'h \<Rightarrow> 'i" assumes "\<And>x y. f x y = g (x, y)" shows "(\<lambda>(x, y). f x y) = g"
fold_end_sum_chain_fg: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and as :: "'a permutation list"   and s :: "'a permutation"   and bs :: "'a permutation list" assumes "(f, g) \<in> foldpairs"   and "as \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "sum_list as `\<rightarrow> C0 \<in> f \<turnstile> \<C>"   and "sum_list (as @ [s]) `\<rightarrow> C0 \<in> g \<turnstile> \<C>"   and "bs \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "induced_automorph f g ` sum_list (as @ [s] @ bs) `\<rightarrow> C0 = sum_list (as @ bs) `\<rightarrow> C0"
array_set_dyn_def: fixes dflt :: "'b"   and a :: "'b array"   and i :: "nat"   and v :: "'b" shows "array_set_dyn dflt a i v \<equiv> Array.len a \<bind> (\<lambda>l. if i < l then Array.upd i v a else let ns = max (2 * l) (i + 1) in array_grow a ns dflt \<bind> Array.upd i v)"
lhs_step_RES: fixes X :: "'b set"   and m :: "'b nres" assumes "\<And>x. x \<in> X \<Longrightarrow> RETURN x \<le> m" shows "RES X \<le> m"
Cons_4: shows "lst (s, x # xs) = fst ss2"
eval_monom_expr: fixes a :: "'a" assumes "a \<in> carrier R" shows "eval R R id a (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<zero>"
execute_parallel_plan_reaches_goal_iff_goal_is_last_element_of_trace: fixes G :: "'b \<Rightarrow> bool option"   and I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" shows "(G \<subseteq>\<^sub>m execute_parallel_plan I \<pi>) = (G \<subseteq>\<^sub>m last (trace_parallel_plan_strips I \<pi>))"
hyps_4: fixes i :: "nat" assumes "i < length qs" shows "accs (reduce_rules \<delta> (f_accessible \<delta> {qs ! i})) (n ! i) (qs ! i)"
finite_empty_induct: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite A"   and "P A"   and "\<And>a A. \<lbrakk>finite A; a \<in> A; P A\<rbrakk> \<Longrightarrow> P (A - {a})" shows "P {}"
Imp_E: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> A"   and "insert B H \<turnstile> C" shows "insert (A IMP B) H \<turnstile> C"
lam_simps_3: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : local.prod a b \<rightarrow> c\<guillemotright>" shows "cod (\<Lambda> a b c f) = exp b c"
degree_interpolation_poly: fixes alg :: "interpolation_algorithm"   and xs_ys :: "('a \<times> 'a) list" shows "degree (interpolation_poly alg xs_ys) \<le> length xs_ys - 1"
vint_def: shows "\<int>\<^sub>\<circ> = ZFC_in_HOL.set (range vint_of_int)"
Crypt_imp_keysFor: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> H"   and "K \<in> symKeys" shows "K \<in> keysFor H"
solution_S2: fixes s :: "'s" assumes "s \<in> S2" shows "x s = 1"
diff_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "minus x y = inf x (uminus y)"
AxCones: fixes m :: "Body"   and k :: "Body"   and x :: "'a Point"   and lightCone :: "Body \<Rightarrow> 'a Point \<Rightarrow> 'a Cone"   and v :: "'a Point" assumes "IOb m"   and "IOb k" shows "onCone x (lightCone m v) \<longrightarrow> onCone (wvt k m x) (lightcone k (wvt k m v))"
standard_decomp_nonempty_unique: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "finite X"   and "valid_decomp X ps"   and "standard_decomp k ps"   and "ps\<^sub>+ \<noteq> []" shows "k = Min (poly_deg ` fst ` set (ps\<^sub>+))"
P1_real_ideal_DDH_advantage_true_bound: fixes \<sigma> :: "bool"   and n :: "nat"   and M :: "'grp \<times> 'grp"   and z :: "'d"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('e \<Rightarrow> 'f spmf)"   and D :: "'f \<times> 'grp \<Rightarrow> bool spmf" assumes "\<sigma> = True" shows "mal_def.adv_P1 n M \<sigma> z (P1_S1 n, P1_S2) \<A> D \<le> local.ddh.advantage n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D) + local.ddh.advantage n (local.ddh.DDH_\<A>' n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D))"
preserves_comp: fixes g :: "'a2"   and f :: "'a2" assumes "A2.seq g f" shows "F2 (g \<cdot>\<^sub>A\<^sub>2 f) = F2 g \<cdot>\<^sub>B\<^sub>2 F2 f"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = []\<^sub>\<circ>" shows "\<FF>\<lparr>ObjMap\<rparr> = []\<^sub>\<circ>"
alwaysI: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "\<And>i. P (\<sigma> |\<^sub>s i)" shows "(\<box>P) \<sigma>"
base\<^sub>L_def: shows "mylens.base\<^sub>L = \<lparr>lens_get = mylens.truncate, lens_put = \<lambda>s v. mylens.extend v (mylens.more s)\<rparr>"
circuit_inD_aux: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "circuit_in \<E> X" shows "indep_system.circuit \<E> (indep_in \<E>) X"
left_unit: fixes a :: "('a \<times> 'a) set" assumes "a \<in> carrier_local_ring_at" shows "add_local_ring_at zero_local_ring_at a = a"
af_subsequence_M_GF_advice: fixes i :: "nat"   and n :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set"   and \<psi> :: "'b ltln" assumes "i \<le> n"   and "suffix n w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> n])[X]\<^sub>\<nu>"   and "\<And>j. j \<le> i \<Longrightarrow> suffix n w \<Turnstile>\<^sub>n af \<psi> (w [j \<rightarrow> n])[X]\<^sub>\<nu>" shows "suffix (Suc n) w \<Turnstile>\<^sub>n af (\<phi> M\<^sub>n \<psi>) (prefix (Suc n) w)[X]\<^sub>\<nu>"
sum_upto_ln_stirling_weak_bigo: shows "(\<lambda>x. sum_upto (\<lambda>x. ln (real x)) x - x * ln x + x) \<in> O(ln)"
pdg_digraphs: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "digraph \<alpha> (\<AA> i)"
kD_but_not_vD_is_iD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> kD rt"   and "ip \<notin> vD rt" shows "ip \<in> iD rt"
sorted_sort_key: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "sorted (map f (sort_key f xs))"
1: shows "Fvars (rawpsubst \<chi> (zip (map fst txs) us)) = Fvars \<chi> - set us \<union> \<Union> {if u \<in> Fvars \<chi> then FvarsT t else {} |t u. (t, u) \<in> set (zip (map fst txs) us)}"
MarkGuardsI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and f :: "'f" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P mark_guards f c Q,A"
relpow_impl: fixes new :: "'a list"   and n :: "nat" shows "set_of (relpow_i new empty n) = {uu_. \<exists>a b m. uu_ = b \<and> a \<in> set new \<and> m \<le> n \<and> (a, b) \<in> set rel ^^ m}"
not_col_permutation_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<not> Col C A B"
fmrestrict_set_insert_notin_dom_m: fixes xa :: "'a"   and N :: "('a, 'b) fmap"   and l1 :: "'a set" assumes "xa \<notin># dom_m N" shows "fmrestrict_set (insert xa l1) N = fmrestrict_set l1 N"
bit_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (\<forall>n. bit a n = bit b n)"
J_rad_def: fixes R :: "('c, 'd) Ring_scheme" shows "J_rad R = (if zeroring R then carrier R else \<Inter> {mx. maximal_ideal R mx})"
right_zero_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v + 0\<^sub>v n = v"
supp_Alts: fixes e1 :: "exp"   and e2 :: "exp" shows "supp (Alts e1 e2) = supp e1 \<union> supp e2"
Sup_distl: fixes x :: "'b"   and Y :: "'b set" shows "x \<cdot> \<Squnion> Y = \<Squnion> ((\<cdot>) x ` Y)"
max_top2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a" assumes "class.order_top less_eq less top" shows "ord.max less_eq x top = top"
diagonal_step_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and i :: "nat"   and j :: "nat"   and d :: "'b"   and v :: "'b" shows "diagonal_step A i j d v = (\<chi>a b. if a = from_nat i \<and> b = from_nat i then d else if a = from_nat j \<and> b = from_nat j then v * A $h from_nat j $h from_nat j else A $h a $h b)"
dir2: shows "\<forall>s\<^sub>1 s\<^sub>2. S (s\<^sub>1, s\<^sub>2) \<sqsubseteq> (P s\<^sub>1||Q s\<^sub>2)"
lvl_update: fixes x :: "'c"   and y :: "'d"   and t :: "(('c \<times> 'd) \<times> nat) tree" obtains "lvl (update x y t) = lvl t"   | "lvl (update x y t) = lvl t + 1" and "sngl (update x y t)"
add: fixes b1 :: "'c"   and b2 :: "'c" shows "g (b1 + b2) = g b1 + g b2"
expectation_gpv'_def: shows "expectation_gpv' \<equiv> expectation_gpv"
scc_of_empty_conv: fixes u :: "'a" shows "(G.scc_of u = {}) = (u \<notin> verts G)"
carrier_irr_def: shows "carrier_irr = {x. degree x < degree f}"
pre_hensel_3: fixes n :: "nat" shows "val_Zp (f' \<bullet> ns n) = val_Zp (f' \<bullet> a)"
index_first: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i < index xs x" shows "x \<noteq> xs ! i"
content_doublesplit: fixes e :: "real"   and k :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "real" assumes "0 < e"   and "k \<in> Basis" obtains d :: "real" where "0 < d" and "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e"
max_node_def: fixes l :: "('b \<times> 'c \<times> 'b) list" shows "max_node l \<equiv> Max (insert (0::'b) (fst ` set l \<union> (snd \<circ> snd) ` set l))"
l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes heap_is_wellformed parent_child_rel \<equiv> heap_is_wellformed = l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes get_disconnected_nodes \<and> parent_child_rel = l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_parent_child_rel get_child_nodes"
chamber_distance_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexIsomorphism X Y f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D" shows "ChamberComplex.chamber_distance Y (f ` C) (f ` D) = ChamberComplex.chamber_distance X C D"
op_list_append_elem_def: fixes s :: "'c list"   and x :: "'c" shows "op_list_append_elem s x \<equiv> s @ [x]"
irrefl: fixes x :: "'a" assumes "x \<in> A" shows "\<not> x \<bowtie> x"
max_Var_floatariths_append: fixes xs :: "floatarith list"   and ys :: "floatarith list" shows "max_Var_floatariths (xs @ ys) = max (max_Var_floatariths xs) (max_Var_floatariths ys)"
create_name_igba_def: fixes \<phi> :: "'a ltlr" shows "create_name_igba \<phi> \<equiv> create_name_gba \<phi> \<bind> (\<lambda>A. gba_to_idx A \<bind> (\<lambda>A'. stat_set_data_nres (card (g_V A)) (card (g_V0 A')) (igbg_num_acc A') \<bind> (\<lambda>_. RETURN A')))"
star_numeral_def: fixes k :: "num" shows "numeral k = star_of (numeral k)"
degree_inverse_irred_power_poly_le: fixes c :: "'a"   and n :: "nat" shows "degree (inverse_irred_power_poly c n) \<le> n - 1"
vwalk_join_def: fixes p :: "'a list"   and q :: "'a list" shows "p \<oplus> q \<equiv> p @ tl q"
IH: fixes y :: "'a list"   and k :: "nat" assumes "length y < length xs" shows "T_brauns k y = 4 * length y"
pareto_efficient': fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "pareto_efficient_scf agents alts scf"   and "pref_profile_wf agents alts R"   and "x \<prec>[Pareto R] y" shows "x \<notin> scf R"
square_part_even_power': fixes x :: "'b"   and n :: "nat" shows "square_part (x ^ (2 * n)) = normalize (x ^ n)"
mon_pl_set: fixes w :: "('d set \<times> 'd set) list" shows "mon_pl w = \<Union> {fst e \<union> snd e |e. e \<in> set w}"
minus_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y \<oplus> x = \<zero>"   and "x \<oplus> y' = \<zero>"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "y' \<in> carrier G" shows "y = y'"
id': shows "real_of_1 (p, l', r') = real_of_1 (p, l, r)"
atom_map_atom_pres: fixes x :: "'a" assumes "atom x" shows "atom_map x = \<eta> x"
rel_cases: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R3 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and a :: "('g, 'i, 'k) generat"   and b :: "('h, 'j, 'l) generat" assumes "rel_generat R1 R2 R3 a b" obtains x :: "'g" and y :: "'h" where "a = Pure x" and "b = Pure y" and "R1 x y"   | x1a :: "'i" and x2 :: "'k" and y1a :: "'j" and y2 :: "'l" where "a = IO x1a x2" and "b = IO y1a y2" and "R2 x1a y1a" and "R3 x2 y2"
disjoint_iff_not_equal: fixes A :: "'c set"   and B :: "'c set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
d2_def: shows "st.d2 = prod_list st.dims2"
set_zip_conv: fixes A :: "'c list"   and B :: "'b list" shows "(set (zip A B))\<inverse> = set (zip B A)"
even_mult_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (times a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
OK_def: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c program" shows "OK I F = (\<forall>i\<in>I. \<forall>j\<in>I - {i}. Acts (F i) \<subseteq> AllowedActs (F j))"