hcju/isabelleps huggingface dataset:
complex_derivative_transform_within_open: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "f holomorphic_on s"   and "g holomorphic_on s"   and "open s"   and "z \<in> s"   and "\<And>w. w \<in> s \<Longrightarrow> f w = g w" shows "deriv f z = deriv g z"
lookup_higher: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and u :: "'a"   and v :: "'a" shows "lookup (punit.higher p u) v = (if u \<prec> v then lookup p v else (0::'c))"
simps_18: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f4 :: "('b, 'c) Combinators \<Rightarrow> ('b, 'c) Combinators \<Rightarrow> 'a"   and x31 :: "'b"   and x32 :: "'b"   and x33 :: "'c" shows "(case AllowPortFromTo x31 x32 x33 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f3 x31 x32 x33"
Lcm_mod_ring_def: shows "Lcm = normalization_euclidean_semiring_class.Lcm"
"3_3": assumes "x \<noteq> y"   and "\<not> rel x y"   and "transp rel"   and "\<And>x y. x \<noteq> y \<Longrightarrow> rel x y \<or> rel y x"   and "sorted_wrt rel (x # xs)"   and "sorted_wrt rel ys" shows "sorted_wrt rel (merge_wrt rel (x # xs) ys)"
intros_2: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
elims_2: fixes x :: "'a list option"   and xa :: "'a list option" assumes "proper_interval x xa" obtains "x = None" and "xa = None"   | xs :: "'a list" where "x = None" and "xa = Some xsa__" and "xsa__ \<noteq> []"   | xs :: "'a list" where "x = Some xsb__" and "xa = None"   | xs :: "'a list" and ys :: "'a list" where "x = Some xs" and "xa = Some ys" and "local.proper_interval_list_aux xs ys"
lazy_channel_insec'_def: shows "lazy_channel_insec' = \<dagger>local.lazy_channel_insec"
consistent_one_step_axioms: shows "consistent_one_step t0 (t0 + e) solution (euler_increment f) 1 (euler_C TYPE('a)) r B'"
5: shows "coreflexive ((d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top) * top * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)\<^sup>T)"
linorder_cases: fixes x :: "'a word"   and y :: "'a word" obtains "x <s y"   | "x = y"   | "y <s x"
assms: shows "ordered_sel_resolvent_upon Sel P1 P2 C A"
sym_iff_eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "sympartp (\<sqsubseteq>) x y = (x = y \<and> y \<sqsubseteq> y)"
that: fixes v :: "'a llist"   and a :: "'a" assumes "w = v $ <a>" shows "thesis"
2138: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - (x \<squnion> y) = top"
inf_aci_1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
Int_UN_distrib2: fixes A :: "'b \<Rightarrow> 'a set"   and I :: "'b set"   and B :: "'c \<Rightarrow> 'a set"   and J :: "'c set" shows "\<Union> (A ` I) \<inter> \<Union> (B ` J) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
dims_nth_lt_n: fixes l :: "nat" assumes "l < n" shows "dims ! l = 2"
assms_2: shows "comm_monoid_mult_class.coprime b c"
subseqs_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "subseqs (map f xs) = map (map f) (subseqs xs)"
ag_r_inv1: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e" assumes "aGroup A"   and "x \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> x = \<zero>\<^bsub>A\<^esub>"
simps_3: fixes \<A> :: "'c ltl set"   and \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "\<A> \<Turnstile>\<^sub>P \<phi> and \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<and> \<A> \<Turnstile>\<^sub>P \<psi>)"
BigAnd_semantics: fixes A :: "'b \<Rightarrow> bool"   and F :: "'b formula list" shows "A \<Turnstile> \<^bold>\<And>F = (\<forall>f\<in>set F. A \<Turnstile> f)"
min_dist_is_dist: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and v' :: "nat" assumes "Graph.connected c v v'" shows "Graph.dist c v (Graph.min_dist c v v') v'"
one_im_sheaf_def: shows "imS.one_im_sheaf \<equiv> \<lambda>V. \<one>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
nonstop_rgt_ex: fixes m :: "nat"   and lm :: "'b"   and stpa :: "nat" assumes "nonstop m (bl2wc (<lm>)) stpa = 0" shows "\<exists>r. conf m (bl2wc (<lm>)) stpa = trpl 0 0 r"
design_iso_points_index_eq: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>" shows "\<B> index ps = \<B>' index \<pi> ` ps"
Collect_ex_eq: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
pos_def: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and S :: "'b set"   and rs :: "'c stream"   and r :: "'c" assumes "RuleSystem eff rules S" shows "RuleSystem.pos rs r \<equiv> LEAST n. shd (sdrop n rs) = r"
wf_dom_def: fixes G :: "nat graph_rec"   and ss :: "nat list list"   and w :: "nat list" assumes "dom_sl G" shows "dom_sl.wf_dom G ss w \<equiv> (\<forall>\<tau>\<in>set ss. \<tau> \<in> (rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G))) \<and> (\<forall>p<length (cfg_doms.nodes G). Sorted_Less.sorted (rev (ss ! p)) \<and> (ss ! p \<noteq> rev [0..<length (cfg_doms.nodes G)] \<longrightarrow> (\<forall>x\<in>set (ss ! p). x < p)) \<and> (ss ! p = rev [0..<length (cfg_doms.nodes G)] \<longrightarrow> (\<exists>x\<in>set w. (x, p) \<in> g_E G \<and> x < p)) \<and> (p \<notin> set w \<longrightarrow> stable nodes_le (cfg_doms.exec G) ss p)) \<and> Sorted_Less.sorted w \<and> length ss = length (cfg_doms.nodes G) \<and> (\<forall>x\<in>set w. x < length (cfg_doms.nodes G))"
sorted_ssort': fixes empty :: "'a"   and is_empty :: "'a \<Rightarrow> bool"   and of_list :: "'b list \<Rightarrow> 'a"   and multiset :: "'a \<Rightarrow> 'b multiset"   and remove_max :: "'a \<Rightarrow> 'b \<times> 'a"   and inv :: "'a \<Rightarrow> bool"   and l :: "'a"   and sl :: "'b list" assumes "RemoveMax empty is_empty of_list multiset remove_max inv"   and "inv l"   and "sorted sl \<and> (\<forall>x\<in>Collection.set multiset l. \<forall>y\<in>set sl. x \<le> y)" shows "sorted (RemoveMax.ssort' is_empty remove_max l sl)"
compute_flow_val_def: fixes am :: "nat \<Rightarrow> nat list"   and cf :: "nat \<times> nat \<Rightarrow> int" shows "compute_flow_val am cf \<equiv> am_get am s \<bind> (\<lambda>succs. nfoldli succs (\<lambda>_. True) (\<lambda>x a. cf_get cf (x, s) \<bind> (\<lambda>b. return (a + b))) 0)"
reverse_subpaths_join: fixes p :: "real \<Rightarrow> 'a" shows "subpath 1 (1 / 2) p +++ subpath (1 / 2) 0 p = reversepath p"
CallUndefined: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and s :: "'d" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile>Call p \<down> Normal s"
CompDiag_Diag_Ide: fixes t :: "'c local.term"   and a :: "'c local.term" assumes "Diag t"   and "Ide a"   and "Dom t = Cod a" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> a = t"
r_into_rtranclp: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "r x y" shows "r\<^sup>*\<^sup>* x y"
singular_subdivision_0: fixes p :: "nat" shows "singular_subdivision p 0 = 0"
equalityI: fixes A :: "'f set"   and B :: "'f set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
sum_distrib_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and r :: "'c" assumes "class.semiring_0 plus zero times" shows "times (comm_monoid_add.sum plus zero f A) r = comm_monoid_add.sum plus zero (\<lambda>n. times (f n) r) A"
Sigma_mset_empty2: fixes A :: "'a multiset" shows "A \<times># {#} = {#}"
inducts: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and xa :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "orbitp f x xa"   and "P (f x)"   and "\<And>y. \<lbrakk>orbitp f x y; P y\<rbrakk> \<Longrightarrow> P (f y)" shows "P xa"
a_zdz: fixes m :: "int"   and n :: "int" shows "ant m - ant n = ant (m - n)"
rref_condition4_explicit: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'f" assumes "reduced_row_echelon_form A"   and "\<not> is_zero_row i A"   and "i \<noteq> j" shows "A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'d)) = (0::'d)"
obtains_dbm_ge: fixes x :: "'a"   and m :: "int" assumes "x \<in> X"   and "m \<le> int (k x)" obtains M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" where "{u \<in> V. real_of_int m \<le> u x} = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n" and "normalized M"
4: shows "\<turnstile> concl (Thm (Imp p (Imp p p)))"
continuous_within_exp: fixes z :: "'d"   and s :: "'d set" shows "continuous (at z within s) exp"
assms_3: shows "\<forall>n. n \<in># N \<longrightarrow> n = 1"
ResB_2: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"   and "y \<noteq> a"   and "y \<noteq> x"   and "y \<noteq> u" shows "<\<nu>y>P \<Longrightarrow>\<^sub>lu in <\<nu>y>P''\<rightarrow>a<x> \<prec> <\<nu>y>P'"
keys_swap\<^sub>0: fixes b :: "nat"   and i :: "nat"   and mp :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "keys (swapPoly\<^sub>0 b i mp) = swap0 b i ` keys mp"
fTo_conditional_min2: fixes C :: "'a set"   and Ca :: "'a set set" assumes "C \<in> carrier fTo D"   and "Chain (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca"   and "Ca = {}" shows "minimum_elem (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) (upper_bounds (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca) C"
P0: shows "\<forall>m k. N < m \<and> N < k \<longrightarrow> eint (int n) < val_Zp_dist (s m) (s k)"
elims_3: fixes x :: "'a"   and xa :: "'b list"   and xb :: "'a" assumes "\<not> cas x xa xb" obtains u :: "'a" and v :: "'a" where "x = ua__" and "xa = []" and "xb = va__" and "ua__ \<noteq> va__"   | u :: "'a" and e :: "'b" and es :: "'b list" and v :: "'a" where "x = u" and "xa = e # es" and "xb = v" and "\<not> (tail G e = u \<and> cas (head G e) es v)"
i_complex_of_real: fixes r :: "real" shows "\<i> * cor r = Complex 0 r"
In0_def: fixes M :: "('a, 'b) node set" shows "In0 M \<equiv> Scons (Numb 0) M"
pow_mining_suc: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "hmining t (Suc n)" shows "PoW t n < PoW t (Suc n)"
smult_one: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<one>\<^bsub>K\<^esub> \<odot>\<^bsub>V\<^esub> x = x"
map_simp_ide: fixes a :: "'a" assumes "ide a" shows "\<ll> a = \<l>[a]"
next_subseqs_spec_1: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and head :: "'d"   and tail :: "'d iarray"   and prevs :: "(nat \<times> 'c) list"   and out :: "'c list"   and head' :: "'d"   and tail' :: "'d iarray"   and prevs' :: "(nat \<times> 'c) list" assumes "my_subseqs.next_subseqs f (head, tail, prevs) = (out, head', tail', prevs')" shows "set prevs' = {uu_. \<exists>v i j. uu_ = (j, f (tail !! j) v) \<and> (i, v) \<in> set prevs \<and> j < i}"
AllN_1: fixes z :: "nat"   and L :: "real list" assumes "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "var + z < length L"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) d = Dv" shows "eval F (L[var + z := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub_fm_helper var a b c d F z) (L[var + z := sqrt Cv])"
residues_mult_nat_def: fixes n :: "nat" shows "residues_mult_nat n \<equiv> 1 < n"
cond_nsg: fixes H :: "'a set" assumes "G \<guillemotright> H "   and "\<forall>a\<in>carrier G. \<forall>h\<in>H. a \<cdot> h \<cdot> \<rho> a \<in> H" shows "G \<triangleright> H"
diff_closed: fixes g :: "'r"   and h :: "'r" assumes "g \<in> R"   and "h \<in> R" shows "g - h \<in> R"
assms_1: shows "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"
last_token_of_path: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and q :: "(symbol \<times> character list) list"   and Doc :: "character list"   and k :: "nat"   and u :: "nat"   and q' :: "(symbol \<times> character list) list"   and t :: "symbol \<times> character list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "q \<in> LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u"   and "q = q' @ [t]"   and "charslength q' = k" shows "t \<in> LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u"
select_convs_1: fixes xpos :: "nat"   and ypos :: "nat"   and more :: "'a" shows "xpos \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr> = xpos"
h1: fixes x :: "'a" shows "h (1, x) = f' x"
mergesort_by_rel_merge_induct: fixes P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" assumes "\<And>xs. P xs []"   and "\<And>ys. P [] ys"   and "\<And>x xs y ys. \<lbrakk>R x y; P xs (y # ys)\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>x xs y ys. \<lbrakk>\<not> R x y; P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
exphr_def: fixes x :: "real" shows "exphr x \<equiv> st (sumhr (0, whn, \<lambda>n. inverse (fact n) * x ^ n))"
transfer: shows "rel_fun cr_gen_ssa_cfg_wf (=) gen_inEdges gen_ssa_wf_inEdges"
IH: fixes \<phi> :: "'fmla"   and s :: "'trm"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s"   and "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t"   and "distinct (map snd txs)"   and "s \<in> trm"   and "y \<in> var"   and "y \<notin> snd ` set txs" shows "rawpsubst (subst \<phi> s y) txs = rawpsubst \<phi> (txs @ [(s, y)])"
\<R>_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains "(case \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   | "(case \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   | "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
gb_schema_aux_term_wf: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "dickson_grading d" shows "wf (gb_schema_aux_term d gs)"
finite_fv\<^sub>s\<^sub>t: fixes S :: "('d, 'c) strand_step list" shows "finite (fv\<^sub>s\<^sub>t S)"
carrier_vec_conjugate: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "conjugate v \<in> carrier_vec n"
m_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier G2"   and "y \<in> carrier G2" shows "x \<otimes>\<^bsub>G2\<^esub> y \<in> carrier G2"
transfer_bounded_cbilinear_bounded_clinearI: fixes g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'c" assumes "g = (\<lambda>i. cblinfun_apply (f i))" shows "bounded_cbilinear g = bounded_clinear f"
mult_imp_div_pos_le: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less_eq x (times z y)" shows "less_eq (divide x y) z"
Im_Arccos_of_real: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "Im (Arccos (complex_of_real x)) = 0"
axioms_2: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset" assumes "sub_incidence_system \<U> \<A> \<V> \<B>" shows "incidence_system \<U> \<A>"
1: shows "\<l>[f \<star> g] \<cdot> \<a>[trg f, f, g] = \<l>[f] \<star> g"
inner_mult_right': fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'b"   and b :: "'b"   and m :: "real" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner a (b * of_real m) = inner a b * m"
in_dgrad_p_set_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "(p \<in> dgrad_p_set d m) = (\<forall>v\<in>keys p. d (pp_of_term v) \<le> m)"
simps_1: fixes n :: "nat"   and w :: "'a vec" shows "adjuster n w [] = 0\<^sub>v n"
init: fixes Mkt :: "('b, 'c) discrete_market"   and v :: "real"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and asset :: "'c" shows "remaining_qty Mkt v pf asset 0 = (\<lambda>w. 0)"
declared_not_undeclared: fixes G :: "prog"   and m :: "memberdecl"   and C :: "qtname" assumes "G\<turnstile> m declared_in C" shows "\<not> G\<turnstile> memberid m undeclared_in C"
iter: fixes g :: "'a \<Rightarrow> 'a"   and n :: "nat" shows "\<forall>a. iter g (g a) n = g (iter g a n)"
log_le_zero_cancel_iff: fixes a :: "real"   and x :: "real" assumes "1 < a"   and "0 < x" shows "(log a x \<le> 0) = (x \<le> 1)"
div_self: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide a a = one"
\<pi>_ln_\<pi>_asymptotics: shows "(\<lambda>x. \<pi> x * ln (\<pi> x)) \<sim>[at_top] (\<lambda>x. x)"
sndE: fixes P :: "'b ptrm"   and Y :: "'b ptrm" assumes "PSnd P \<approx> Y" obtains B :: "'b ptrm" where "Y = PSnd B" and "P \<approx> B"
zip_option_intros_1: fixes ys :: "'c list"   and xs :: "'d list"   and zs :: "('d \<times> 'c) list" assumes "length ys = length xs"   and "length zs = length xs"   and "zs = zip xs ys" shows "zip_option xs ys = Some zs"
shift_eq_scons: fixes u :: "'c list"   and v :: "'c stream"   and a :: "'c"   and w :: "'c stream" shows "(u @- v = a ## w) = (u = [] \<and> v = a ## w \<or> (\<exists>u'. u = a # u' \<and> u' @- v = w))"
simplicial_simplex_def: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "simplicial_simplex p S f \<equiv> singular_simplex p (subtopology (powertop_real UNIV) S) f \<and> (\<exists>l. f = oriented_simplex p l)"
total_var: fixes x :: "'a" shows "times_top_class.total x = total_var x"
q': shows "(red F)\<^sup>*\<^sup>* q q'"
of_seq_snth: fixes xs :: "'b stream" shows "of_seq ((!!) xs) = xs"
relpowp_E2: fixes n :: "nat"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and z :: "'c" assumes "(P ^^ n) x z" obtains "n = 0" and "x = z"   | y :: "'c" and m :: "nat" where "n = Suc m" and "P x y" and "(P ^^ m) y z"
kruskal_acyclic_inv: fixes f :: "'a"   and q :: "'a"   and e :: "'a" assumes "pd_kleene_allegory_class.acyclic f"   and "covector q"   and "(f \<sqinter> q)\<^sup>T * f\<^sup>\<star> * e = bot"   and "e * f\<^sup>\<star> * e = bot"   and "forest_components f \<le> - e" shows "pd_kleene_allegory_class.acyclic (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
k_empt_def: shows "SPRsingle.KBP.k_empt \<equiv> \<lparr>aActs = MapOps.empty trie_odlist_MapOps, aTrans = MapOps.empty trans_MapOps\<rparr>"
sxalloc_no_jump: fixes G :: "prog"   and s0 :: "state"   and s1 :: "state"   and j :: "jump" assumes "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)" shows "fst s1 \<noteq> Some (Jump j)"
distproj_self_2: fixes x :: "'c"   and A :: "'c set" assumes "x \<in> A" shows "distproj x A = x"
induct: fixes P :: "ipt_l4_ports list \<Rightarrow> bool"   and a0 :: "ipt_l4_ports list" assumes "P []"   and "\<And>proto ps. P [L4Ports proto ps]"   and "\<And>proto1 ps1 proto2 ps2 pss. (\<not> proto1 \<noteq> proto2 \<Longrightarrow> P (L4Ports proto1 (wi2l (wordinterval_intersection (l2wi ps1) (l2wi ps2))) # pss)) \<Longrightarrow> P (L4Ports proto1 ps1 # L4Ports proto2 ps2 # pss)" shows "P a0"
shuffle_terms_distinct_iff: fixes p :: "('a list \<times> 'b) list"   and s :: "('a list \<times> 'b) list" assumes "map (\<lambda>(a, y). (mset a, y)) p = map (\<lambda>(a, y). (mset a, y)) s" shows "(\<forall>x\<in>set p. distinct (fst x)) = (\<forall>x\<in>set s. distinct (fst x))"
Sup_upper: fixes Inf :: "'f set \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and bot :: "'f"   and top :: "'f"   and x :: "'f"   and A :: "'f set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
Arr_comp: fixes f :: "('a E.term, 'a) arr"   and g :: "('a E.term, 'a) arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "g \<cdot> f \<noteq> Null \<and> Dom (g \<cdot> f) \<in> IDE \<and> Cod (g \<cdot> f) \<in> IDE \<and> Map (g \<cdot> f) \<in> HOM (Dom (g \<cdot> f)) (Cod (g \<cdot> f))"
n2m_ctor_fold_G_def: fixes s1 :: "('e, 'g) G0 \<Rightarrow> 'f"   and s2 :: "('f, 'g) F0 \<Rightarrow> 'g" shows "n2m_ctor_fold_G s1 s2 = ctor_fold_G (s1 \<circ> map_pre_G\<^sub>M id id (ctor_fold_F (s2 \<circ> BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf)) \<circ> BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf)"
ParTR_MtransC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c2 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c1 :: "('a, 'c) com" assumes "PL.MtransT tval aval (c2, s) s'" shows "PL.MtransC tval aval (c1 | c2, s) (c1, s')"
Max_eq_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(linorder.Max (\<sqsubseteq>) A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<sqsubseteq> m))"