hcju/isabelleps huggingface dataset:
Hensel_next_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and t :: "'e"   and R' :: "('e set, 'h) Ring_scheme"   and Y :: "'e set"   and f :: "'e"   and m :: "nat"   and gh :: "'e \<times> 'e" shows "Hen\<^bsub> R S X t R' Y f\<^esub> m gh = (SOME gh1. gh1 \<in> carrier R \<times> carrier R \<and> deg R S X (fst gh1) \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh1)) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (fst gh \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst gh1) \<and> deg R S X (snd gh1) + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh1)) \<le> deg R S X f \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (snd gh \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> snd gh1) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst gh1 \<cdot>\<^sub>r\<^bsub>R\<^esub> snd gh1))"
cyclic_on_def: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set" shows "cyclic_on f S = (\<exists>s\<in>S. S = orbit f s)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R S.map \<rr>"
fold_A_im_min_gallery: fixes Cs :: "'a set list" assumes "SimplicialComplex.min_maxsimpchain (fold_A \<turnstile> A) (C # Cs)" shows "min_gallery (C # Cs)"
uint_range_size: fixes w :: "'a word" shows "0 \<le> uint w \<and> uint w < 2 ^ size w"
neq_def: fixes p :: "tm" shows "neq p = not (eq p)"
empty: fixes g :: "'d \<Rightarrow> 'c" shows "prod g {} = (1::'c)"
absolutely_integrable_on_subcbox: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "f absolutely_integrable_on S"   and "cbox a b \<subseteq> S" shows "f absolutely_integrable_on cbox a b"
f_inv_def: fixes I :: "nat \<Rightarrow> bool"   and n :: "nat" shows "f_inv I n \<equiv> f n"
wls_psubstAbs_idEnv: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (us, s) A" shows "A $[idEnv] = A"
assms_1: shows "(f, g) \<in> fundfoldpairs"
Union_rel_empty: fixes R :: "('c \<times> 'd set) set" shows "([], op_empty_coll) \<in> clw_rel R"
distinct_40: fixes x51 :: "CTypeId"   and x52 :: "nat" shows "nullV \<noteq> objV x51 x52"
ORLEFT_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set"   and Y :: "ntrm set" shows "pi \<bullet> ORLEFT (A OR B) X Y = ORLEFT (A OR B) (pi \<bullet> X) (pi \<bullet> Y)"
multi_member_last: fixes x :: "'f" shows "x \<in># {#x#}"
G_def: shows "G \<equiv> {x. x \<bullet> n = g}"
assms_7: shows "limsup (\<lambda>x. ereal (aa (x + 1) / prod aa {0..x} powr (2 + 2 / \<epsilon> + \<delta>) * (1 / bb (x + 1)))) = \<infinity>"
pred42E: fixes n :: "nat" assumes "pred n = 42" obtains "n = 43"
smc_SemiCAT_obj_initialD: fixes \<AA> :: "V" assumes "obj_initial (smc_SemiCAT \<alpha>) \<AA>" shows "\<AA> = smc_0"
step_1: shows "LCons v (LCons w P'') = P'"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "\<rho>.map f = null"
g_2: shows "agree_on (LG e\<^sub>1 v) f g"
prime_elem_normalize_iff: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "comm_semiring_1.prime_elem times one zero (normalize x) = comm_semiring_1.prime_elem times one zero x"
stabiliser_def: fixes x :: "'b" shows "stabiliser x = {g \<in> carrier G. g \<odot> x = x}"
inf_prefix_length_Suc: fixes a :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "inf_prefix_length (a # xs) (a # ys) = Suc (inf_prefix_length xs ys)"
path_component_of_subset_topspace: fixes X :: "'a topology"   and x :: "'a" shows "path_component_of_set X x \<subseteq> topspace X"
Nil: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile> \<langle>[],[],s\<rangle> \<Rightarrow> s"
irreflpI: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a. \<not> R a a" shows "irreflp R"
hsize_chop: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "Suc (hsize (chop t)) = hsize t"
hyps: shows "1 + real n * x \<le> (1 + x) ^ n"
rel_interior_closure_convex_shrink: fixes S :: "'a set"   and c :: "'a"   and x :: "'a"   and e :: "real" assumes "convex S"   and "c \<in> rel_interior S"   and "x \<in> closure S"   and "0 < e"   and "e \<le> 1" shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
word_le_minus_mono_right: fixes z :: "'b word"   and y :: "'b word"   and x :: "'b word" assumes "z \<le> y"   and "y \<le> x"   and "z \<le> x" shows "x - y \<le> x - z"
smap_less_def: fixes m1 :: "'c \<Rightarrow> 'd set"   and m2 :: "'c \<Rightarrow> 'd set" shows "(m1 \<subseteq>. m2) = (\<forall>k. m1 k \<subseteq> m2 k)"
llist_of_list_of: fixes xs :: "'b llist" assumes "lfinite xs" shows "llist_of (list_of xs) = xs"
encode_operator_precondition_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and op :: "'b strips_operator" shows "encode_operator_precondition \<Pi> t op \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O> in \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ops op))) \<^bold>\<or> Atom (State t (index vs v))) (precondition_of op)"
prems_1: shows "j \<le> mod_type_class.from_nat (Suc k)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M" shows "l_delete_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
simps_2: fixes \<O> :: "nat set"   and r :: "'b memref"   and rs :: "'b memref list" shows "weak_sharing_consistent \<O> (r # rs) = (case r of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then L \<subseteq> A \<and> A \<inter> R = {} \<and> R \<subseteq> \<O> \<and> weak_sharing_consistent (\<O> \<union> A - R) rs else weak_sharing_consistent \<O> rs | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> L \<subseteq> A \<and> A \<inter> R = {} \<and> R \<subseteq> \<O> \<and> weak_sharing_consistent (\<O> \<union> A - R) rs | _ \<Rightarrow> weak_sharing_consistent \<O> rs)"
apx_subset: fixes Z :: "('c \<Rightarrow> real) set" shows "Z \<subseteq> Approx\<^sub>\<beta> Z"
p_Mp: fixes f :: "int poly" shows "p.eq_m (Mp f) f"
p_2: shows "prime (Ring_Divisibility.mult_of R) p"
is_chain_extend: fixes S :: "(nat, 'b) form set"   and C :: "(nat, 'b) form set set"   and f :: "nat \<Rightarrow> (nat, 'b) form" shows "is_chain (extend S C f)"
coprime_ord: fixes n :: "nat"   and a :: "nat" assumes "coprime n a" shows "0 < ord n a \<and> [a ^ ord n a = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < ord n a \<longrightarrow> [a ^ m \<noteq> 1] (mod n))"
bounded_linear_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'd" shows "bounded_linear ((*\<^sub>v) a)"
set_mset_eq_empty_iff: fixes M :: "'b multiset" shows "(set_mset M = {}) = (M = {#})"
collapse: fixes rule :: "'b rule" shows "Rule (get_match rule) (get_action rule) = rule"
ntrp_add_context_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list el_step"   and s' :: "'n list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrp fg"   and "mon_w fg (le_rem_s e) \<inter> mon_c fg cn = {}" shows "((s, c + cn), e, s', c' + cn) \<in> ntrp fg"
inject: fixes a :: "'a"   and b :: "'c"   and a' :: "'a"   and b' :: "'c" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
elims_1: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and x :: "'a"   and xa :: "enat list"   and xb :: "'a"   and y :: "bool" assumes "WGraph.path w x xa xb = y" obtains u :: "'a" and v :: "'a" where "x = ua__" and "xa = []" and "xb = v__" and "y = (ua__ = v__)"   | u :: "'a" and l :: "enat" and ls :: "enat list" and v :: "'a" where "x = u" and "xa = l # ls" and "xb = v" and "y = (\<exists>uh. l = w (u, uh) \<and> WGraph.path w uh ls v)"
init_fields_def: fixes FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" shows "init_fields FDTs \<equiv> (map_of \<circ> map (\<lambda>((F, D), b, T). ((F, D), default_val T))) FDTs"
store_buffer_step_induct: fixes x1a :: "nat \<Rightarrow> nat"   and x1b :: "'b memref list"   and x1c :: "'c"   and x1d :: "'d"   and x1e :: "'e"   and x2a :: "nat \<Rightarrow> nat"   and x2b :: "'b memref list"   and x2c :: "'c"   and x2d :: "'d"   and x2e :: "'e"   and P :: "(nat \<Rightarrow> nat) \<Rightarrow> 'b memref list \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> 'b memref list \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" assumes "(x1a, x1b, x1c, x1d, x1e) \<rightarrow>\<^sub>w (x2a, x2b, x2c, x2d, x2e)"   and "\<And>m volatile a aa b v A L R W rs \<O> \<R> \<S>. P m (Write\<^sub>s\<^sub>b volatile a (aa, b) v A L R W # rs) \<O> \<R> \<S> (m(a := v)) rs \<O> \<R> \<S>" shows "P x1a x1b x1c x1d x1e x2a x2b x2c x2d x2e"
exhaust: fixes y :: "'a simplex_state" obtains x :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" where "y = Simplex_State x"
c: shows "list_all2 (\<approx>) (map f y) (map f' ya)"
fun_mult_closed: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "g \<in> carrier (function_ring (carrier Zp) Zp)" shows "f \<otimes>\<^bsub>function_ring (carrier Zp) Zp\<^esub> g \<in> carrier (function_ring (carrier Zp) Zp)"
Rmi_sv_2: fixes s :: "bdd"   and ni :: "nat"   and n :: "nat ifex"   and ni' :: "nat"   and n' :: "nat ifex" assumes "bdd_sane s"   and "(ni, n) \<in> Rmi s"   and "(ni', n') \<in> Rmi s"   and "ni \<noteq> ni'" shows "n \<noteq> n'"
principal_char_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" shows "principal_char G a = (if a \<in> carrier G then 1 else 0)"
mult_not_zero: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
exhaust: fixes y :: "inferences" obtains "y = Ground"   | "y = FirstOrder"
lt_ex: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.no_bot less_eq less" shows "\<exists>y. less y x"
sg6: shows "stop t = [] \<or> stop t = [stop_vc]"
I0: shows "I {} S\<^sub>0 \<sigma>\<^sub>0"
qPsubstBinp_def: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" shows "qbinp %%[[qrho]] \<equiv> lift (qPsubstAbs qrho) qbinp"
distl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<squnion> z) = x \<cdot> y \<squnion> x \<cdot> z"
order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> U"   and "y \<in> U"   and "z \<in> U"   and "x \<le>\<^sub>o\<^sub>w y"   and "y \<le>\<^sub>o\<^sub>w z" shows "x \<le>\<^sub>o\<^sub>w z"
1: shows "(\<Union>i. A i \<inter> b) = \<Union> (range A) \<inter> b"
heq_4: shows "map of_int_hom.vec_hom fs' ! i = map of_int_hom.vec_hom fs ! (i - 1)"
sorted_def: fixes m :: "nat"   and t :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "LoadBalancing m" shows "LoadBalancing.sorted t j = (\<forall>x\<in>{1..j}. \<forall>y\<in>{1..x}. t x \<le> t y)"
Infinitesimal_star_of_mult2: fixes x :: "'a star"   and r :: "'a" assumes "x \<in> Infinitesimal" shows "star_of r * x \<in> Infinitesimal"
lim_imp_Liminf: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and f0 :: "'c" assumes "F \<noteq> \<bottom>"   and "(f \<longlongrightarrow> f0) F" shows "Liminf F f = f0"
generate_idem'_right: fixes G :: "('c, 'd) monoid_scheme"   and A :: "'c set"   and B :: "'c set" assumes "Group.group G"   and "A \<subseteq> carrier G"   and "B \<subseteq> carrier G" shows "generate G (A \<union> generate G B) = generate G (A \<union> B)"
Z_def: shows "Z = {x @\<^sub>v vec_of_scal (0::'a) |x. x \<in> X}"
comp_char: fixes g :: "'a"   and f :: "'a" shows "R.comp g f = (if R.arr f \<and> R.arr g \<and> seq g f then g \<cdot> f else null)"
unital_quantale_homset_def: shows "unital_quantale_homset = {f. comp_pres f \<and> Sup_pres f \<and> un_pres f}"
\<ll>_ide_simp: fixes a :: "'a" assumes "S.ide a" shows "M.\<ll> a = M.lunit a"
distributed_integrable: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "distributed M N X (\<lambda>x. ennreal (f x))"   and "g \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x" shows "integrable N (\<lambda>x. f x * g x) = integrable M (\<lambda>x. g (X x))"
cprank_def: fixes A :: "'b tensor" shows "cprank A = (LEAST n. cprank_max n A)"
dom_add_closed: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rdom (rdom x \<union> rdom y) = rdom x \<union> rdom y"
rank_initial: fixes x :: "nat" shows "\<exists>i. rank x x = Some i"
Sup1: fixes x :: "'c Assertion"   and A :: "'c Assertion set" assumes "x \<in> A" shows "x \<le> \<Squnion>A"
unique_factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "q.unique_factorization_m f cfs = (q.Mf ` Collect (q.factorization_m f) = {q.Mf cfs})"
power_decreasing_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero b"   and "less b one" shows "less_eq (power.power one times b m) (power.power one times b n) = (n \<le> m)"
simps_1: fixes f :: "'a" shows "Inv \<^bold>\<langle>f\<^bold>\<rangle> = \<^bold>\<langle>C.inv f\<^bold>\<rangle>"
EX_MR: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and ps :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "applied_rule_rev Cp x p \<noteq> \<lfloor>DenyAll\<rfloor>"   and "p = DenyAll # ps" shows "applied_rule_rev Cp x p = applied_rule_rev Cp x ps"
h13: shows "[(ainv * b') ^ (4 * k) * ((ainv * b')\<^sup>2 + 1) = 1 * ((ainv * b')\<^sup>2 + 1)] (mod 4 * k + 3)"
tf: fixes i :: "nat" assumes "i < n" shows "Bij_Nat.to_nat (Bij_Nat.from_nat i) = i"
Quotient3_tmap_Abs_Rep: fixes R1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs1 :: "'g \<Rightarrow> 'h"   and Rep1 :: "'h \<Rightarrow> 'g"   and R2 :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and Abs2 :: "'i \<Rightarrow> 'j"   and Rep2 :: "'j \<Rightarrow> 'i"   and ts :: "('h, 'j) tllist" assumes "Quotient3 R1 Abs1 Rep1"   and "Quotient3 R2 Abs2 Rep2" shows "tmap Abs1 Abs2 (tmap Rep1 Rep2 ts) = ts"
g0: fixes x :: "real" assumes "0 < x"   and "x < a" shows "(g0 has_real_derivative g' x) (at x)"
ptensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "a.ptensor_mat m1 m2 = tensor_mat a.dims0 a.vars1' m1 m2"
trop_ueval: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'e"   and x :: "('g, 'f) uexpr"   and y :: "('h, 'f) uexpr"   and z :: "('i, 'f) uexpr"   and b :: "'f" shows "\<lbrakk>trop f x y z\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b) (\<lbrakk>y\<rbrakk>\<^sub>e b) (\<lbrakk>z\<rbrakk>\<^sub>e b)"
eventually_gt_at_top: fixes c :: "'c" shows "eventually ((<) c) at_top"
induct: fixes x1 :: "trm"   and x2 :: "fin_string \<Rightarrow> real"   and x3 :: "real"   and P :: "trm \<Rightarrow> (fin_string \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool" assumes "[x1]x2 \<down> x3"   and "\<And>w r \<nu>. Rep_bword w \<equiv>\<^sub>E r \<Longrightarrow> P (Const w) \<nu> r"   and "\<And>x \<nu>. P (Var x) \<nu> (\<nu> x)"   and "\<And>\<theta>\<^sub>1 \<nu> r1 \<theta>\<^sub>2 r2. \<lbrakk>[\<theta>\<^sub>1]\<nu> \<down> r1; P \<theta>\<^sub>1 \<nu> r1; [\<theta>\<^sub>2]\<nu> \<down> r2; P \<theta>\<^sub>2 \<nu> r2\<rbrakk> \<Longrightarrow> P (Plus \<theta>\<^sub>1 \<theta>\<^sub>2) \<nu> (r1 + r2)"   and "\<And>\<theta>\<^sub>1 \<nu> r1 \<theta>\<^sub>2 r2. \<lbrakk>[\<theta>\<^sub>1]\<nu> \<down> r1; P \<theta>\<^sub>1 \<nu> r1; [\<theta>\<^sub>2]\<nu> \<down> r2; P \<theta>\<^sub>2 \<nu> r2\<rbrakk> \<Longrightarrow> P (Times \<theta>\<^sub>1 \<theta>\<^sub>2) \<nu> (r1 * r2)"   and "\<And>\<theta>\<^sub>1 \<nu> r1 \<theta>\<^sub>2 r2. \<lbrakk>[\<theta>\<^sub>1]\<nu> \<down> r1; P \<theta>\<^sub>1 \<nu> r1; [\<theta>\<^sub>2]\<nu> \<down> r2; P \<theta>\<^sub>2 \<nu> r2\<rbrakk> \<Longrightarrow> P (trm.Max \<theta>\<^sub>1 \<theta>\<^sub>2) \<nu> (max r1 r2)"   and "\<And>\<theta>\<^sub>1 \<nu> r1 \<theta>\<^sub>2 r2. \<lbrakk>[\<theta>\<^sub>1]\<nu> \<down> r1; P \<theta>\<^sub>1 \<nu> r1; [\<theta>\<^sub>2]\<nu> \<down> r2; P \<theta>\<^sub>2 \<nu> r2\<rbrakk> \<Longrightarrow> P (trm.Min \<theta>\<^sub>1 \<theta>\<^sub>2) \<nu> (min r1 r2)"   and "\<And>\<theta>\<^sub>1 \<nu> r1. \<lbrakk>[\<theta>\<^sub>1]\<nu> \<down> r1; P \<theta>\<^sub>1 \<nu> r1\<rbrakk> \<Longrightarrow> P (Abs \<theta>\<^sub>1) \<nu> \<bar>r1\<bar>"   and "\<And>\<theta> \<nu> r. \<lbrakk>[\<theta>]\<nu> \<down> r; P \<theta> \<nu> r\<rbrakk> \<Longrightarrow> P (Neg \<theta>) \<nu> (- r)" shows "P x1 x2 x3"
unit_part: shows "\<guillemotleft>\<iota> \<star> f : (b \<star> b) \<star> f \<Rightarrow>\<^sub>S b \<star> f\<guillemotright> \<and> S.iso (\<iota> \<star> f)"
weakBisimWeakUpto: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "eqvt X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto><(weakBisim O X O bisim)> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<approx> Q"
v_minus_p_orthogonal_complement: fixes S :: "'b set"   and X :: "'b set"   and v :: "'b" assumes "subspace S"   and "independent X"   and "X \<subseteq> S"   and "S \<subseteq> span X"   and "pairwise orthogonal X" shows "v - proj_onto v X \<in> orthogonal_complement S"
noneg: shows "\<not> has_disc_negated is_Src_Ports False m"
lub_eqI: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c set"   and l :: "'c" assumes "class.po below"   and "po.is_lub below M l" shows "po.lub below M = l"
take_indices_lem_R: fixes n :: "nat"   and arb_list :: "('a list \<times> 'a list) list"   and index_list :: "nat list" assumes "n < length (take_indices arb_list index_list)"   and "\<forall>q. List.member index_list q \<longrightarrow> q < length arb_list" shows "\<exists>k<length arb_list. take_indices arb_list index_list ! n = arb_list ! k"
ccompatible_def: fixes \<R> :: "('a \<Rightarrow> 'b) set set"   and cc :: "('a, 'b) cconstraint" shows "ccompatible \<R> cc \<equiv> \<forall>R\<in>\<R>. R \<subseteq> Regions.ccval cc \<or> Regions.ccval cc \<inter> R = {}"
group_hom: shows "group_hom G (BijGroup E) \<phi>"
ibounded_by_def: fixes "consider" :: "'call \<Rightarrow> bool" shows "ibounded_by consider = (\<lambda>(\<A>1, \<A>2) q. \<exists>q1 q2. (\<forall>pk. interaction_bounded_by consider (\<A>1 pk) q1) \<and> (\<forall>cipher \<sigma>. interaction_bounded_by consider (\<A>2 cipher \<sigma>) q2) \<and> q1 + q2 \<le> q)"
stuck_def: fixes e :: "exp" shows "stuck e \<equiv> \<nexists>e'. e \<longrightarrow> e'"
matrix_left_invertible_span_rows_gen: fixes A :: "(('a, 'c) vec, 'b) vec" shows "(\<exists>B. B ** A = mat (1::'a)) = (vec.span (rows A) = UNIV)"
image_Un: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'b set" shows "f ` (A \<union> B) = f ` A \<union> f ` B"
WT_parallel_oracle: fixes \<I>l :: "('a, 'b) \<I>"   and left :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c) spmf"   and sl :: "'c"   and \<I>r :: "('d, 'e) \<I>"   and right :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf"   and sr :: "'f" assumes "\<I>l \<turnstile>c left sl \<surd>"   and "\<I>r \<turnstile>c right sr \<surd>" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c parallel_oracle left right (sl, sr) \<surd>"
indexed_pmult_zero: fixes i :: "'c" shows "indexed_const \<zero> \<Otimes> i = indexed_const \<zero>"
summable_iff_convergent': fixes f :: "nat \<Rightarrow> 'e" shows "summable f = convergent (\<lambda>n. sum f {..n})"
rep_fset_def: shows "rep_fset \<equiv> quot_type.rep Rep_fset"
Zr_times: fixes n :: "int"   and m :: "int" shows "n \<cdot>\<^sub>r\<^bsub>Zr\<^esub> m = n * m"
bi_unique_rel_set_lemma: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and X :: "'e set"   and Y :: "'f set" assumes "bi_unique R"   and "rel_set R X Y" obtains f :: "'e \<Rightarrow> 'f" where "Y = f ` X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
intro: fixes E :: "('c \<times> 'c) set"   and S :: "'c set" assumes "E \<subseteq> S \<times> S"   and "\<And>a. a \<in> S \<Longrightarrow> (a, a) \<in> E"   and "\<And>a b. (a, b) \<in> E \<Longrightarrow> (b, a) \<in> E"   and "\<And>a b c. \<lbrakk>(a, b) \<in> E; (b, c) \<in> E\<rbrakk> \<Longrightarrow> (a, c) \<in> E" shows "equivalence S E"
mult_single: fixes k :: "'b"   and a :: "'c"   and l :: "'b"   and b :: "'c" shows "monomial a k * monomial b l = monomial (a * b) (k + l)"
two_block_non_interfering_row_comp1: fixes i :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "i < dim_row A" shows "row (two_block_non_interfering A B) i = row A i @\<^sub>v 0\<^sub>v (dim_col B)"
diamondp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "diamondp r \<equiv> \<forall>x y1 y2. r x y1 \<longrightarrow> r x y2 \<longrightarrow> (\<exists>z. r y1 z \<and> r y2 z)"