hcju/isabelleps huggingface dataset:
smult_carrier_vec: fixes a :: "'c"   and v :: "'c vec"   and n :: "nat" shows "(a \<cdot>\<^sub>v v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
shadow_root_eq2: fixes ptr' :: "(_) element_ptr" shows "|h \<turnstile> get_shadow_root ptr'|\<^sub>r = |h' \<turnstile> get_shadow_root ptr'|\<^sub>r"
br'_invar_QcD: fixes Q :: "'d set"   and W :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set" assumes "(Q, W, rcm) \<in> br'_invar \<delta>" shows "{lhs r |r. r \<in> \<delta> \<and> set (rhsq r) \<subseteq> Q - W} \<subseteq> Q"
approximating_imp_booloan_semantics_nospoofing: fixes \<gamma> :: "'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and rs :: "'a common_primitive rule list"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and \<Gamma> :: "char list \<Rightarrow> 'a common_primitive rule list option" assumes "matcher_agree_on_exact_matches (\<gamma>::'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool) common_matcher"   and "simple_ruleset (rs::'a common_primitive rule list)"   and "no_spoofing TYPE('b) (ipassmt::iface \<Rightarrow> ('a word \<times> nat) list option) (rs::'a common_primitive rule list)" shows "\<forall>iface::iface\<in>dom (ipassmt::iface \<Rightarrow> ('a word \<times> nat) list option). \<forall>p::('a, 'b) tagged_packet_scheme. (\<Gamma>::char list \<Rightarrow> 'a common_primitive rule list option),(\<gamma>::'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool),p\<lparr>p_iiface := iface_sel iface\<rparr>\<turnstile> \<langle>(rs::'a common_primitive rule list), Undecided\<rangle> \<Rightarrow> Decision FinalAllow \<longrightarrow> p_src p \<in> ipcidr_union_set (set (the (ipassmt iface)))"
f_def: fixes ix :: "'a \<times> 'b" shows "f ix = (case ix of (i, x) \<Rightarrow> x)"
lookup': fixes as :: "'a option list" shows "Mapping.lookup tuple_in' as = safe_max (fst ` {tas \<in> ts_tuple_rel (set (linearize data_in')). valid_tuple tuple_since tas \<and> as = snd tas})"
swf_def: fixes f :: "('a \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> alt" shows "swf f \<equiv> \<lambda>P. {(a, b). a \<noteq> b \<and> f (Top {a, b} \<circ> P) = b}"
list_emb_code_1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list" shows "list_emb P [] ys = True"
par_skip: fixes a :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<parallel> skip = a"
trd_red_rtrancl: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "(red (set fs))\<^sup>*\<^sup>* p (trd fs p)"
type_update_select_conv_select_return_top: fixes cts :: "ct list" assumes "ct_suffix [TSome T_i32] cts"   and "3 \<le> length cts" shows "type_update_select (TopType cts) = select_return_top cts (cts ! (length cts - 2)) (cts ! (length cts - 3))"
igSwap_termMOD: shows "igSwap termMOD = swap"
hyps: fixes C :: "'a set" assumes "C \<in> f \<turnstile> \<C>"   and "D \<in> \<C> - f \<turnstile> \<C>"   and "gallery (C # Es @ [D])" shows "\<exists>As A B Bs. A \<in> f \<turnstile> \<C> \<and> B \<in> \<C> - f \<turnstile> \<C> \<and> C # Es @ [D] = As @ A # B # Bs"
nb\<^sub>2: shows "v \<in> {v \<in> set (\<Pi>\<^sub>\<V>). (\<Pi>\<^sub>G) v \<noteq> None}"
set_cases: fixes e :: "'c"   and a :: "'c list" assumes "e \<in> set a" obtains z2 :: "'c list" where "a = e # z2__"   | z1 :: "'c" and z2 :: "'c list" where "a = z1 # z2" and "e \<in> set z2"
SORTED_L: shows "sorted_wrt_dist p (nearest_nbors n ps p l)"
dghm_up_ArrMap_vdomain: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" shows "\<D>\<^sub>\<circ> (dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
weak_transition_stepI: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>"   and "weak_nominal_ts.observable_transition transition \<tau> P \<alpha> P'" shows "weak_nominal_ts.weak_transition transition \<tau> P \<alpha> P'"
IH: assumes "odd x"   and "1 < x" shows "Jacobi 2 x = (- 1) ^ (((nat x)\<^sup>2 - 1) div 8)"
Greatest_equality: fixes P :: "'b word \<Rightarrow> bool"   and x :: "'b word" assumes "P x"   and "\<And>y. P y \<Longrightarrow> y \<le>s x" shows "signed.Greatest P = x"
contractible_space: fixes X :: "'b topology" shows "contractible_space X = (topspace X = {} \<or> (\<exists>a\<in>topspace X. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)))"
euclidean_size_of_nat: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (semiring_1.of_nat one plus zero n) = n"
a: shows "\<Gamma>,\<gamma>,p\<lparr>p_iiface := iface_sel iface\<rparr>\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow"
env_restr_below_trans: fixes S1 :: "'a set"   and m1 :: "'a \<Rightarrow> 'b"   and m2 :: "'a \<Rightarrow> 'b"   and S2 :: "'a set"   and m3 :: "'a \<Rightarrow> 'b" assumes "m1 f|` S1 \<sqsubseteq> m2 f|` S1"   and "m2 f|` S2 \<sqsubseteq> m3 f|` S2" shows "m1 f|` (S1 \<inter> S2) \<sqsubseteq> m3 f|` (S1 \<inter> S2)"
mset_pos_as_neg: fixes M :: "'a zmultiset" shows "zmset_of (mset_pos M) = zmset_of (mset_neg M) + M"
fBall_pred_weaken: fixes M :: "'a fset"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. \<lbrakk>x |\<in>| M; P x\<rbrakk> \<Longrightarrow> Q x"   and "fBall M P" shows "fBall M Q"
encode_problem_parallel_correct_iii_2: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat"   and op :: "'b strips_operator"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k < length (\<Phi>\<inverse> \<Pi> \<A> t)"   and "op \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)" shows "v \<in> set (delete_effects_of op) \<longrightarrow> (\<Phi>\<^sub>S\<inverse> \<Pi> \<A> Suc k) v = Some False"
coprime_cong_mult_factorization_poly: fixes P :: "'b poly set"   and a :: "'b poly"   and b :: "'b poly" assumes "finite P"   and "P \<subseteq> Collect irreducible"   and "Ball P (cong a b)"   and "\<forall>p1 p2. p1 \<in> P \<and> p2 \<in> P \<and> p1 \<noteq> p2 \<longrightarrow> comm_monoid_mult_class.coprime p1 p2" shows "[a = b] (mod \<Prod>P)"
that_2: shows "(g has_derivative g') (at x within {x..})"
rel_refl_strong: fixes x :: "'d \<Rightarrow> 'c"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> range x \<Longrightarrow> Ra z z" shows "rel_fun (=) Ra x x"
Fall2_def: fixes p :: "pvar"   and p' :: "pvar"   and \<phi> :: "'a dfmla" shows "Fall2 p p' \<phi> \<equiv> Fall p (Fall p' \<phi>)"
nz: fixes x :: "'b"   and y :: "'b" assumes "x \<in> A"   and "y \<in> A"   and "x \<noteq> y"   and "h x = h y" shows "g (h x) = \<^bold>1"
Posix_elims_5: fixes s :: "'b list"   and r1 :: "'b rexp"   and r2 :: "'b rexp"   and v :: "'b val" assumes "s \<in> Times r1 r2 \<rightarrow> v" obtains s1 :: "'b list" and v1 :: "'b val" and s2 :: "'b list" and v2 :: "'b val" where "s = s1 @ s2" and "v = Seq v1 v2" and "s1 \<in> r1 \<rightarrow> v1" and "s2 \<in> r2 \<rightarrow> v2" and "\<forall>s\<^sub>3. s1 @ s\<^sub>3 \<in> lang r1 \<longrightarrow> s\<^sub>3 = [] \<or> (\<forall>s\<^sub>4. s\<^sub>3 @ s\<^sub>4 = s2 \<longrightarrow> s\<^sub>4 \<notin> lang r2)"
wfactors_perm_cong_l: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c"   and fs' :: "'c list" assumes "Group.comm_monoid G"   and "wfactors G fs a"   and "fs <~~> fs'"   and "set fs \<subseteq> carrier G" shows "wfactors G fs' a"
sterms_pinduct: fixes a0 :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and P :: "('e \<Rightarrow> ('f, 'g, 'e, 'h) seqp) \<Rightarrow> ('f, 'g, 'e, 'h) seqp \<Rightarrow> bool"   and a1 :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed a0"   and "\<And>\<Gamma> p1 p2. \<lbrakk>sterms_dom (\<Gamma>, p1 \<oplus> p2); P \<Gamma> p1; P \<Gamma> p2\<rbrakk> \<Longrightarrow> P \<Gamma> (p1 \<oplus> p2)"   and "\<And>\<Gamma> pn. \<lbrakk>sterms_dom (\<Gamma>, call(pn)); P \<Gamma> (\<Gamma> pn)\<rbrakk> \<Longrightarrow> P \<Gamma> (call(pn))"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}\<langle>va\<rangle> vb) \<Longrightarrow> P \<Gamma> ({v}\<langle>va\<rangle> vb)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}\<lbrakk>va\<rbrakk> vb) \<Longrightarrow> P \<Gamma> ({v}\<lbrakk>va\<rbrakk> vb)"   and "\<And>\<Gamma> v va vb vc vd. sterms_dom (\<Gamma>, {v}unicast(va, vb) . vc \<triangleright> vd) \<Longrightarrow> P \<Gamma> ({v}unicast(va, vb) . vc \<triangleright> vd)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}broadcast(va) . vb) \<Longrightarrow> P \<Gamma> ({v}broadcast(va) . vb)"   and "\<And>\<Gamma> v va vb vc. sterms_dom (\<Gamma>, {v}groupcast(va, vb) . vc) \<Longrightarrow> P \<Gamma> ({v}groupcast(va, vb) . vc)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}send(va) . vb) \<Longrightarrow> P \<Gamma> ({v}send(va) . vb)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}deliver(va) . vb) \<Longrightarrow> P \<Gamma> ({v}deliver(va) . vb)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}receive(va) . vb) \<Longrightarrow> P \<Gamma> ({v}receive(va) . vb)" shows "P a0 a1"
prod_in_Rats: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<rat>" shows "prod f A \<in> \<rat>"
hom_sum: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set" shows "[:sum f X:] = (\<Sum>x\<in>X. [:f x:])"
attractive_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "attractive B (\<sqsubseteq>)"
tauLaw3SimLeft: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a prefix" assumes "eqvt Rel"   and "(\<Psi>, P, Q) \<in> Rel"   and "\<And>xvec Tvec. length xvec = length Tvec \<Longrightarrow> (\<Psi>, P[xvec::=Tvec], Q[xvec::=Tvec]) \<in> Rel"   and "\<And>\<Psi> P Q R S. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> Rel; \<Psi> \<rhd> R \<sim> S\<rbrakk> \<Longrightarrow> (\<Psi>, P, S) \<in> Rel"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> Rel \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<leadsto><Rel> \<alpha>\<cdot>Q"
absorb_iff2: fixes b :: "'d multiset"   and a :: "'d multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"
standard_identity_rules_3: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph" assumes "standard' C G" shows "maintained (symmetry_rule S_Idt) G"
ground_sat_n: shows "lifted_q_calc.ground.saturated (lifted_q_calc.\<G>_Fset N)"
wf_induct: fixes r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "wf r"   and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
leq_refl: fixes x :: "'c nres" shows "x \<le> x"
expandExecution: fixes exec :: "('p, 'v, 's) configuration list"   and msg :: "('p, 'v) message"   and cfg :: "('p, 'v, 's) configuration"   and trace :: "('p, 'v) message list" assumes "stepReachable (last exec) msg cfg"   and "execution trans sends start exec trace" shows "\<exists>exec' trace'. execution trans sends start exec' trace' \<and> prefixList exec exec' \<and> prefixList trace trace' \<and> last exec' = cfg \<and> msg \<in> set (drop (length trace) trace')"
in_hhomI: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.src \<mu> = a"   and "Span.trg \<mu> = b" shows "Span.in_hhom \<mu> a b"
composable_implies_arr_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "arr \<nu>"
isomorphic_implies_hpar_3: fixes f :: "'b"   and f' :: "'b" assumes "f \<cong>\<^sub>D f'" shows "src\<^sub>D f = src\<^sub>D f'"
set_merge: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "set_tree (merge t1 t2) = set_tree t1 \<union> set_tree t2"
sp: shows "ps (S t i) (occurs_on (t ! i)) \<noteq> None"
pair_prob_space_def: fixes M1 :: "'b measure"   and M2 :: "'c measure" shows "pair_prob_space M1 M2 \<equiv> pair_sigma_finite M1 M2 \<and> prob_space M1 \<and> prob_space M2"
Fvars_fmla_simps_7: fixes a :: "name"   and A :: "fmla" shows "Fvars (exi a A) = Fvars A - {a}"
base_in_hom_2: shows "\<guillemotleft>t : t \<Rightarrow> t\<guillemotright>"
degree_m_le: fixes f :: "int poly" shows "degree_m f \<le> degree f"
cases_scheme: fixes r :: "'a bw_point_scheme" obtains cx :: "int" and cy :: "int" and black :: "bool" and more :: "'a" where "r = \<lparr>cx = cx, cy = cy, black = black, \<dots> = more\<rparr>"
max_divisors: fixes x :: "nat"   and r :: "nat" assumes "Suc 0 < x" shows "Max {u. x ^ u dvd x ^ r} = r"
matches_tuple: fixes \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'a match_expr"   and a :: "action"   and p :: "'b" shows "matches (\<beta>, \<alpha>) m a p = ternary_to_bool_unknown_match_tac \<alpha> a p (ternary_ternary_eval (map_match_tac \<beta> p m))"
lift_is_unique: fixes f :: "'b \<Rightarrow> bool"   and gens :: "'b set"   and h :: "(bool \<times> 'b) list \<Rightarrow> bool" assumes "Group.group C2"   and "f \<in> gens \<rightarrow> carrier C2"   and "h \<in> hom \<F>\<^bsub>gens\<^esub> C2"   and "\<forall>g\<in>gens. h (\<iota> g) = f g" shows "\<forall>x\<in>carrier \<F>\<^bsub>gens\<^esub>. h x = C2.lift f x"
foo1: fixes i :: "int"   and s :: "'a \<Rightarrow> 'b" shows "s\<langle>a := i\<rangle>\<cdot>a = i"
mult_less_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
simps_3: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and t1 :: "trm"   and t2 :: "trm" shows "sat \<sigma> V v i (formula.Eq t1 t2) = (eval_trm v t1 = eval_trm v t2)"
iso_lunit: fixes a :: "'c" assumes "C.ide a" shows "C.iso \<l>\<^sub>C[a]"
oalist_inv_rawD1: fixes xs :: "('a \<times> 'd) list" assumes "oalist_inv_raw xs" shows "(0::'d) \<notin> snd ` set xs"
assms_3: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "R x y"   and "P xs (y # ys)" shows "P (x # xs) (y # ys)"
iso_sym: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "Group.group G"   and "G \<cong> H" shows "H \<cong> G"
that: fixes w :: "'b stream" assumes "run A (w ||| smap (r \<circ> Suc) nats) (r 0)" shows "thesis"
intros_1: fixes r :: "('g \<times> 'h) set" shows "([], []) \<in> listrel r"
injection_eq: fixes f :: "'d \<rightarrow> 'c"   and g :: "'c \<rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x" shows "(g\<cdot>x = g\<cdot>y) = (x = y)"
pwd_spy_iii_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_iii n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_iii_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
prefix_to_wordset_subset_ipset_from_cidr: fixes pfx :: "'b prefix_match" shows "prefix_to_wordset pfx \<subseteq> ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
Collect_conv_if2: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "{x. a = x \<and> P x} = (if P a then {a} else {})"
toSet_fromList_set: fixes xs :: "'c list" shows "toSet (fromList xs) = set xs"
prems: shows "i # is \<lhd> dims (subtensor_combine ds As)"
DFS_axioms: shows "DFS (subset_succs A) (nfa_is_node A) (subset_invariant A) subset_ins subset_memb subset_empt"
genset_uminus: fixes a :: "'w" assumes "a \<in> S" shows "- a = a"
poly_root_factor_2: fixes q :: "'b poly"   and r :: "'b"   and k :: "'b" shows "(poly (q * [:r, 1::'b:]) k = (0::'b)) = (k = - r \<or> poly q k = (0::'b))"
pdom_pdom_res: fixes A :: "'a set"   and f :: "('a, 'b) pfun" shows "pdom (A \<lhd>\<^sub>p f) = A \<inter> pdom f"
synthetic_div_0: fixes c :: "'a" shows "synthetic_div 0 c = 0"
dghm_tdghm_comp_NTMap_vdomain: fixes \<HH> :: "V" shows "\<D>\<^sub>\<circ> ((\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>)\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
f10: fixes ss :: "symbol list"   and n :: "nat" shows "length ss \<le> n \<or> length (take n ss) = n"
pSwap: shows "presSwapAll (rec MOD) (recAbs MOD) MOD"
norm_one: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool" assumes "class.real_normed_algebra_1 plus zero minus uminus scaleR times one dist norm sgn uniformity open" shows "norm one = 1"
nil_reduced_word_for_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set" assumes "class.monoid_add plus zero" shows "monoid_add.reduced_word_for plus zero A zero []"
derivg: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(g has_real_derivative g' x) (at x within {a..b})"
harmonic_aux3: fixes M :: "nat" shows "1 + real M / 2 \<le> (\<Sum>n = 1..2 ^ M. 1 / real n)"
assms_5: shows "g \<in> punit.reduced_GB (monomial (1::'a) ` S)"
Un_Diff: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<union> B - C = A - C \<union> (B - C)"
Inf_Sup: fixes A :: "'b set"   and sup :: "'b" assumes "is_Inf {b. \<forall>a\<in>A. a \<sqsubseteq> b} sup" shows "is_Sup A sup"
step_def: fixes g :: "'g" shows "i.step g \<equiv> \<lambda>(ua, pa). (CFG_SSA_Transformed_notriv_base.uses' \<alpha>n defs (uses(g := u, g := ua)) (phis(g := p, g := pa)) chooseNext_all g, CFG_SSA_Transformed_notriv_base.phis' \<alpha>n defs (uses(g := u, g := ua)) (phis(g := p, g := pa)) chooseNext_all g)"
arr_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" shows "arr (mkArr A B F) = (A \<subseteq> Univ \<and> |A| <o \<AA> \<and> B \<subseteq> Univ \<and> |B| <o \<AA> \<and> F \<in> A \<rightarrow> B)"
coeff_mult: fixes p :: "'a Polynomial.poly"   and q :: "'a Polynomial.poly"   and n :: "nat" shows "poly.coeff (p * q) n = (\<Sum>i\<le>n. poly.coeff p i * poly.coeff q (n - i))"
card_of_Un_diff_infinite: fixes A :: "'a set"   and B :: "'a set" assumes "infinite A"   and "|B| <o |A|" shows "|A - B| =o |A|"
A: shows "g_min m P = iterate_sel_no_map (iterateoi m) P"
le_fin_trivial: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"
partition_basic: fixes l :: "nat"   and r :: "nat"   and xs :: "'b list"   and rs :: "nat"   and xs' :: "'b list" assumes "l < r"   and "r < length xs"   and "(rs, xs') = Quicksort.partition xs l r" shows "outer_remains xs xs' l r \<and> mset xs' = mset xs \<and> l \<le> rs \<and> rs \<le> r"
behaves_def: shows "L2.behaves = load2 OO L2.sem_behaves"
bind_commute: fixes m :: "'m"   and m' :: "'m"   and f :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m' (f x)) = bind m' (\<lambda>y. bind m (\<lambda>x. f x y))"
wiringI: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and cnv :: "('a, 'b, 'c, 'd) converter"   and f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"   and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>" shows "wiring \<I> \<I>' cnv (f, g)"
loeb_aux_prv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv (\<phi>L \<phi>)" shows "prv \<phi>"
list_set_rel_def: fixes R :: "('g \<times> 'h) set" shows "\<langle>R\<rangle>list_set_rel \<equiv> \<langle>R\<rangle>list_rel O br set distinct"
irreducible\<^sub>d_factor: fixes p :: "'c poly" assumes "0 < degree p" shows "\<exists>q r. irreducible\<^sub>d q \<and> p = q * r \<and> degree r < degree p"
Fun_pr1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "Fun (hfsetcat.pr1 a b) = (\<lambda>x\<^bold>\<in>local.set (hfsetcat.prod a b). UP (hfst (DOWN x)))"
prod_int_eq: fixes i :: "nat"   and j :: "nat" shows "prod int {i..j} = \<Prod>{int i..int j}"
ae2ce_to_env_restr: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ae2ce ae = (\<lambda>_. many) f|` edom ae"
count_mset_set_le_one: fixes A :: "'b set"   and x :: "'b" shows "count (mset_set A) x \<le> 1"
assms_1: shows "redundant C (inferred_clause_sets R S N)"
update_convs_1: fixes wc' :: "int \<Rightarrow> int"   and xc :: "int"   and yc :: "int"   and zc :: "int"   and wc :: "int"   and more :: "'a" shows "wc_update wc' \<lparr>xc = xc, yc = yc, zc = zc, wc = wc, \<dots> = more\<rparr> = \<lparr>xc = xc, yc = yc, zc = zc, wc = wc' wc, \<dots> = more\<rparr>"
learnLessState_def: fixes learnL :: "(Literal list list \<times> Literal list list) set" shows "learnLessState learnL \<equiv> {(stateA, stateB). getM stateA = getM stateB \<and> (getF stateA, getF stateB) \<in> learnL}"
gen_ideal_def: fixes x :: "'a" shows "\<langle>x\<rangle> \<equiv> {y. \<exists>r\<in>R. y = r \<cdot> x}"
the_gpv_parametric': fixes A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (rel_gpv'' A C R) (rel_spmf (rel_generat A C (rel_fun R (rel_gpv'' A C R)))) the_gpv the_gpv"
op_set_cart_def: fixes x :: "'a set"   and y :: "'b set" shows "op_set_cart x y \<equiv> x \<times> y"
Rep_cfun_strict1: fixes x :: "'c" shows "\<bottom>\<cdot>x = \<bottom>"
add_node_def: fixes v :: "'b"   and G :: "'b list_graph" shows "FiniteListGraph.add_node v G = \<lparr>nodesL = if v \<in> set (nodesL G) then nodesL G else v # nodesL G, edgesL = edgesL G\<rparr>"