hcju/isabelleps huggingface dataset:
holomorphic_starlike_primitive: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and k :: "complex set" assumes "continuous_on S f"   and "starlike S"   and "open S"   and "finite k"   and "\<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x" shows "\<exists>g. \<forall>x\<in>S. (g has_field_derivative f x) (at x)"
gcd_dvd2: fixes a :: "'c"   and b :: "'c" shows "normalization_euclidean_semiring_class.gcd a b dvd b"
mult_FactGroup: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set" shows "(\<otimes>\<^bsub>G Mod N\<^esub>) = (<#>\<^bsub>G\<^esub>)"
simps: fixes r1 :: "rat"   and p2 :: "int poly"   and l2 :: "rat"   and r2 :: "rat" shows "add_rat_1 r1 (p2, l2, r2) = (let p = cf_pos_poly (poly_add_rat r1 p2); (l, r, sr) = tighten_poly_bounds_for_x p 0 (l2 + r1) (r2 + r1) (sgn (ipoly p (r2 + r1))) in (p, l, r))"
path_ipd_swap: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> k \<noteq> local.return"   and "k < n" obtains \<pi>' :: "nat \<Rightarrow> 'n" and m :: "nat" where "is_path \<pi>'" and "\<pi> =\<^bsub>n\<^esub> \<pi>'" and "k < m" and "\<pi>' m = ipd (\<pi>' k)" and "\<forall>l\<in>{k..<m}. \<pi>' l \<noteq> ipd (\<pi>' k)"
if_type': fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and e :: "bexp"   and t :: "bexp set"   and P :: "bexp set"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and c\<^sub>1 :: "(Eg1.var, aexp, bexp) Stmt"   and \<Gamma>' :: "Eg1.var \<Rightarrow> bexp set option"   and \<S>' :: "Eg1.var set \<times> Eg1.var set"   and P' :: "bexp set"   and c\<^sub>2 :: "(Eg1.var, aexp, bexp) Stmt"   and P'' :: "bexp set"   and P''' :: "bexp set" assumes "type_bexpr \<Gamma> e t"   and "pred_entailment P t"   and "has_type \<Gamma> \<S> (add_pred P \<S> e) c\<^sub>1 \<Gamma>' \<S>' P'"   and "has_type \<Gamma> \<S> (add_pred P \<S> (bexp_neg e)) c\<^sub>2 \<Gamma>' \<S>' P''"   and "P''' \<subseteq> P' \<inter> P''" shows "has_type \<Gamma> \<S> P (Stmt.If e c\<^sub>1 c\<^sub>2) \<Gamma>' \<S>' P'''"
simps_1: fixes v1 :: "'d \<times> nat"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.ces v1 [] v2 subs = (v1 = v2 \<or> (v1, v2) \<in> subs\<^sup>+)"
assoc_list_of_array_code: fixes a :: "'a array" shows "assoc_list_of_array a = assoc_list_of_array_code a 0"
intros_1: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "\<phi> xs" shows "ev \<phi> xs"
simpleL_def: shows "simpleL \<equiv> if pderiv p * q \<noteq> 0 \<and> odd (1 - order x q) then if sign_r_pos q x then 1 else - 1 else 0"
assms_1: fixes x :: "'b" shows "monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. F f x)"
treap_ins: fixes t :: "('b \<times> 'c) tree"   and k :: "'b"   and p :: "'c" assumes "treap t" shows "treap (ins k p t)"
A'_def: shows "A' = foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<k])"
implication_frontier_iff_implied_frontier_vacant: fixes c :: "('loc, 't) Propagate.configuration"   and b :: "'t"   and loc :: "'loc" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "worklists_vacant_to c b" shows "(b \<in>\<^sub>A frontier (c_imp c loc)) = (b \<in>\<^sub>A implied_frontier (c_pts c) loc)"
llength_lfilter_ile: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "llength (lfilter P xs) \<le> llength xs"
assms: shows "a<x>.P \<longmapsto> \<alpha> \<prec> P'"
u_3: fixes sa :: "'a" assumes "sa \<in> cball x e" shows "s (flow0 sa (u sa)) = 0"
index_uminus_mat_3: fixes A :: "'a mat" shows "dim_col (- A) = dim_col A"
coerce_functor: shows "COERCE('a\<cdot>'b, 'c\<cdot>'b) = fmap\<cdot>COERCE('a, 'c)"
I1: shows "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
wtl_Cons_not_top: fixes i :: "'b"   and "is" :: "'b list"   and c :: "'a list"   and pc :: "nat"   and s :: "'a" shows "(wtl (i # is) c pc s \<noteq> \<top>) = (wtc c pc s \<noteq> \<top> \<and> s \<noteq> \<top> \<and> wtl is c (pc + 1) (wtc c pc s) \<noteq> \<top>)"
has_field_is_class': fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C has F,b:T in D" shows "is_class P D"
find_index_def: shows "find_index \<equiv> \<lambda>uu uua. rec_list (\<lambda>_. 0) (\<lambda>x xs xsa P. if P x then 0 else xsa P + 1) uua uu"
P: shows "\<forall>f. \<forall>g\<in>set (P f). f = g"
sup_Assertion_def: fixes p :: "'c Assertion"   and q :: "'c Assertion" shows "p \<squnion> q = abs_wpt ({\<cdot> p } \<squnion> {\<cdot> q })"
tl_def: fixes list :: "'d list" shows "tl list = (case list of [] \<Rightarrow> [] | x21 # x22 \<Rightarrow> x22)"
ord_option_simps_2: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option" shows "ord_option ord x None = (x = None)"
lim_h_minus_le_lim_h_plus: fixes e :: "'v \<times> 'v" shows "(\<Squnion>n. h_minus n e) \<le> (\<Squnion>n. h_plus n e)"
CHAR_pos_iff: shows "(0 < CHAR) = (\<exists>n>0. of_nat n = (0::'a))"
that: fixes q :: "nat"   and r :: "real" assumes "t = real q * - T + r"   and "0 \<le> r"   and "r < T" shows "thesis"
diff_delta_arctan_upper_34_def: shows "diff_delta_arctan_upper_34 \<equiv> \<lambda>x. - 256 / ((5 + 105 * x\<^sup>2 + 315 * x ^ 4 + 231 * x ^ 6)\<^sup>2 * (1 + x\<^sup>2))"
SV_inv3_v_round: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and D :: "process \<Rightarrow> val option" shows "{SV_inv3} sv_round r S v D {> SV_inv3}"
ENR_imp_absolute_neighbourhood_retract_UNIV: fixes S :: "'a set"   and S' :: "'b set" assumes "ENR S"   and "S homeomorphic S'" obtains T' :: "'b set" where "open T'" and "S' retract_of T'"
semilat_JVM: fixes wf_mb :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and mxs :: "nat"   and mxl :: "nat" assumes "wf_prog wf_mb P" shows "semilat (JVM_SemiType.sl P mxs mxl)"
simps_3: fixes x3 :: "nat"   and y3 :: "nat" shows "(Nonce x3 = Nonce y3) = (x3 = y3)"
rep_eq: fixes x :: "'e pmf" shows "set_pmf x = {xa. measure_pmf.prob x {xa} \<noteq> 0}"
create_element_is_weakly_dom_component_safe_step: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)|\<^sub>r"   and "ptr \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> create_element document_ptr tag|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
is_rect_list_def: fixes rects :: "'a rectangle list" shows "is_rect_list rects = (\<forall>i<length rects. is_rect (rects ! i))"
elims_3: fixes G :: "('a, 'b) graph"   and x :: "'a"   and xa :: "('a \<times> 'b \<times> 'a) list"   and xb :: "'a" assumes "valid_unMultigraph G"   and "\<not> valid_unMultigraph.is_trail G x xa xb" obtains v :: "'a" and v' :: "'a" where "x = va__" and "xa = []" and "xb = v'__" and "\<not> (va__ = v'__ \<and> v'__ \<in> nodes G)"   | v :: "'a" and v1 :: "'a" and w :: "'b" and v2 :: "'a" and ps :: "('a \<times> 'b \<times> 'a) list" and v' :: "'a" where "x = v" and "xa = (v1, w, v2) # ps" and "xb = v'" and "\<not> (v = v1 \<and> (v1, w, v2) \<in> edges G \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> valid_unMultigraph.is_trail G v2 ps v')"
fps_const_neg: fixes c :: "'b" shows "- fps_const c = fps_const (- c)"
induct: fixes x :: "trm"   and P :: "trm \<Rightarrow> bool" assumes "SNa x"   and "\<And>M. \<lbrakk>\<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> SNa M'; \<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> P M'\<rbrakk> \<Longrightarrow> P M" shows "P x"
psi_mono: fixes i1 :: "'b"   and i2 :: "'b"   and x :: "'b sort_key extended \<times> 'b sort_key extended" assumes "i1 < i2"   and "is_interval x" shows "\<Psi> x i1 < \<Psi> x i2"
Beta_Regions'_axioms: shows "Beta_Regions' X v n not_in_X"
tail_arev: fixes a :: "'a \<times> 'a" assumes "a \<in> arcs (with_proj G)" shows "tail (with_proj G) (rev_G a) = head (with_proj G) a"
HBL1_axioms: shows "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P"
join_option_eq_Some: fixes x :: "'c option option"   and y :: "'c" shows "(join_option x = Some y) = (x = Some (Some y))"
domains_char': fixes f :: "cospan_shape.Arr" shows "partial_magma.domains cospan_shape.comp f = (if f = cospan_shape.Null then {} else {cospan_shape.Dom f})"
eq_eqvt_1: fixes pi :: "(tyvrs \<times> tyvrs) list"   and x :: "'a"   and y :: "'a" shows "pi \<bullet> (x = y) = (pi \<bullet> x = pi \<bullet> y)"
split_sel: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (option = Some (the option) \<longrightarrow> P (f2 (the option))))"
typeof_conf: fixes h :: "'heap"   and v :: "'addr val"   and T :: "ty"   and P :: "'b prog" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
know_composition_neg_ine_s: fixes P :: "specID"   and secret :: "Secrets"   and Q :: "specID"   and PQ :: "specID" assumes "\<not> know P (sKS secret)"   and "\<not> know Q (sKS secret)"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ" shows "\<not> ine PQ (sE secret)"
irreducible_linear_field_poly: fixes b :: "'c"   and a :: "'c" assumes "b \<noteq> (0::'c)" shows "irreducible [:a, b:]"
6: shows "A1 = mat_of_row (Matrix.row A 0)"
simps_7: fixes var :: "nat"   and a :: "atom fm"   and b :: "atom fm" shows "grab_atoms var (And a b) = (case grab_atoms var a of None \<Rightarrow> None | Some al \<Rightarrow> case grab_atoms var b of None \<Rightarrow> None | Some bl \<Rightarrow> Some (al @ bl))"
map_poly_Re_mult: fixes p :: "complex poly"   and q :: "complex poly" assumes "set (coeffs p) \<subseteq> \<real>"   and "set (coeffs q) \<subseteq> \<real>" shows "map_poly Re (p * q) = map_poly Re p * map_poly Re q"
inj_idom_hom_compose: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "inj_idom_hom f"   and "inj_idom_hom g" shows "inj_idom_hom (f \<circ> g)"
is_sig_red_singletonI: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and F :: "('c \<Rightarrow>\<^sub>0 'e) set"   and r :: "'c \<Rightarrow>\<^sub>0 'e" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail F r" obtains f :: "'c \<Rightarrow>\<^sub>0 'e" where "f \<in> F" and "qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail {f} r"
subringE_3: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subring H R" shows "\<one>\<^bsub>R\<^esub> \<in> H"
simps_39: fixes x51 :: "expr"   and x52 :: "stmt"   and x53 :: "stmt"   and x2 :: "expr" shows "If(x51) x52 Else x53 \<noteq> Expr x2"
iso_is_retraction: fixes f :: "'a" assumes "A.iso f" shows "A.retraction f"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "D (b1 + b2) = D b1 + D b2"
finite_sup_closed: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" assumes "weak_upper_semilattice L"   and "finite A"   and "A \<subseteq> carrier L"   and "A \<noteq> {}" shows "\<Squnion>\<^bsub>L\<^esub>A \<in> carrier L"
join_commute: fixes x :: "'b"   and y :: "'b" assumes "compatible x y" shows "x \<squnion> y = y \<squnion> x"
Y: shows "OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>) = OclAllInstances_generic pre_post Type->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. \<lfloor>Type Object\<rfloor>) (mk \<lparr>heap = \<sigma>', assocs = A\<rparr>)"
not_sym: fixes t :: "'a"   and s :: "'a" assumes "t \<noteq> s" shows "s \<noteq> t"
nzero: shows "v \<noteq> 0\<^sub>v m"
l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf get_shadow_root get_shadow_root_locs"
prod_list_coprime_right: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and xs :: "'a list"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>x. x \<in> set xs \<Longrightarrow> algebraic_semidom.coprime times one a x" shows "algebraic_semidom.coprime times one a (monoid_mult.prod_list one times xs)"
mode_update_add_anno: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> Sec option"   and upd :: "'Var ModeUpd" assumes "mds_consistent mds \<Gamma>" shows "mds_consistent (update_modes upd mds) (\<Gamma> \<oplus> upd)"
append_in_lists_conv: fixes xs :: "'c list"   and ys :: "'c list"   and A :: "'c set" shows "(xs @ ys \<in> lists A) = (xs \<in> lists A \<and> ys \<in> lists A)"
not_neutral_contains_not_neutral: fixes g :: "'l \<Rightarrow> 'k"   and A :: "'l set" assumes "sum g A \<noteq> (0::'k)" obtains a :: "'l" where "a \<in> A" and "g a \<noteq> (0::'k)"
map_bind_pmf: fixes f :: "'i \<Rightarrow> 'h"   and M :: "'j pmf"   and g :: "'j \<Rightarrow> 'i pmf" shows "map_pmf f (M \<bind> g) = M \<bind> (\<lambda>x. map_pmf f (g x))"
circline_type_pos_card_eq0: fixes H :: "circline" assumes "0 < circline_type H" shows "circline_set H = {}"
pcs_shift: fixes pc :: "nat"   and n :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" assumes "pc < n" shows "pc \<notin> pcs (shift n xt)"
ds_union: shows "ds \<union> ds' = ds' \<union> (ds \<union> ds')"
face_of_convex_hull_subset: fixes S :: "'b set"   and T :: "'b set" assumes "compact S"   and "T face_of convex hull S" obtains s' :: "'b set" where "s' \<subseteq> S" and "T = convex hull s'"
assms_1: shows "HPhase1or2ReadThen s s' p d r"
Diagonalize_Tensor: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u" shows "\<^bold>\<lfloor>t \<^bold>\<otimes> u\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<otimes> \<^bold>\<lfloor>u\<^bold>\<rfloor>\<^bold>\<rfloor>"
assms: shows "dbm_entry_val u (Some c1) (Some c2) (min a b)"
elims: fixes x :: "int"   and xa :: "int"   and xb :: "nat"   and xc :: "int \<times> int \<times> int"   and y :: "int \<times> int \<times> int" assumes "fast_ppoint_mult x xa xb xc = y" obtains m :: "int" and a :: "int" and n :: "nat" and p :: "int \<times> int \<times> int" where "x = m" and "xa = a" and "xb = n" and "xc = p" and "y = (if n = 0 then (0, 0, 0) else if n mod 2 = 0 then mpdouble m a (fast_ppoint_mult m a (n div 2) p) else mpadd m a p (mpdouble m a (fast_ppoint_mult m a (n div 2) p)))"
NotE: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<not>\<phi> in v]" shows "[\<phi> in v] \<longrightarrow> (\<forall>\<psi>. [\<psi> in v])"
s_App_inv: fixes \<pi> :: "Syntax.term list"   and v\<^sub>1 :: "Syntax.term"   and v\<^sub>2 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e :: "Syntax.term" assumes "\<lless>\<pi>, App v\<^sub>1 v\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<ggreater>" obtains e\<^sub>1' :: "Syntax.term" where "ea__ = App e\<^sub>1' v\<^sub>2" and "\<lless>\<pi>, v\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater>"   | e\<^sub>2' :: "Syntax.term" where "ea__ = App v\<^sub>1 e\<^sub>2'" and "value v\<^sub>1" and "\<lless>\<pi>, v\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>2'\<ggreater>"   | x :: "Syntax.var" and e :: "Syntax.term" where "v\<^sub>1 = Lam x__ eb__" and "\<pi>' = \<pi>" and "ea__ = eb__[v\<^sub>2 / x__]" and "value v\<^sub>2" and "atom x__ \<sharp> (v\<^sub>2, \<pi>)"   | x :: "Syntax.var" and e :: "Syntax.term" where "v\<^sub>1 = Rec x e" and "\<pi>' = \<pi>" and "ea__ = App (e[Rec x e / x]) v\<^sub>2" and "value v\<^sub>2" and "atom x \<sharp> (v\<^sub>2, \<pi>)"
emeasure_lborel_cball_finite: fixes c :: "'c"   and r :: "real" shows "emeasure lborel (cball c r) < \<infinity>"
assms: shows "HT_partial_mods \<pi> vs P c Q"
es: shows "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:]) \<and> length es = n"
fine_imp_le: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list" assumes "fine \<delta> (a, b) D" shows "a \<le> b"
\<tau>move2_CastD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and T :: "ty"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (Cast T e) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
is_lasso_prpl_pre_def: fixes prpl :: "'v list \<times> 'v list" shows "is_lasso_prpl_pre prpl \<equiv> case prpl of (pr, pl) \<Rightarrow> \<exists>v0 va. v0 \<in> V0 \<and> pl \<noteq> [] \<and> path E v0 pr va \<and> path E va pl va"
leq_sum: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.idempotent_semiring one times sup less_eq less bot"   and "\<forall>k. less_eq (f k) (g k)" shows "less_eq (comm_monoid_add.sum sup bot f {k. True}) (comm_monoid_add.sum sup bot g {k. True})"
bra_mat_on_vec: fixes A :: "complex mat"   and v :: "complex vec" assumes "dim_col A = dim_vec v" shows "\<langle>A * |v\<rangle>| = \<langle>|v\<rangle>| * A\<^sup>\<dagger>"
converseI: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
finter_finsert_right: fixes A :: "'a fset"   and a :: "'a"   and B :: "'a fset" shows "A |\<inter>| finsert a B = (if a |\<in>| A then finsert a (A |\<inter>| B) else A |\<inter>| B)"
HMA_det: shows "rel_fun HMA_M (=) Determinant.det Determinants.det"
add_order_preserving: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'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 a :: "'b"   and b :: "'b"   and u :: "'b"   and v :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq a b" shows "less_eq (plus (plus u a) v) (plus (plus u b) v)"
bdd_above_Iic: fixes b :: "'d multiset" shows "subset_mset.bdd_above (subset_mset.atMost b)"
simps: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and a1 :: "'g \<times> 'i"   and a2 :: "'h \<times> 'j" shows "rel_prod R1 R2 a1 a2 = (\<exists>a b c d. a1 = (a, c) \<and> a2 = (b, d) \<and> R1 a b \<and> R2 c d)"
inducts: fixes xa :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "xa \<in> orbit f x"   and "P (f x)"   and "\<And>y. \<lbrakk>y \<in> orbit f x; P y\<rbrakk> \<Longrightarrow> P (f y)" shows "P xa"
monom_mult_zero_left: fixes t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.monom_mult (0::'c) t p = 0"
assms_4: fixes var :: "Vars.var"   and c :: "'a"   and exp :: "exp" assumes "{atom var} \<sharp>* c"   and "\<And>c. P c exp" shows "P c (Lam [var]. exp)"
C3r_var: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y \<cdot> x \<le> y" shows "z \<cdot> x\<^sup>\<star> \<le> y"
least_H_le_least_A: shows "(LEAST n. H $ ia $ n \<noteq> (0::'a)) \<le> (LEAST n. A $ ia $ n \<noteq> (0::'a))"
chosen_def: fixes u :: "'a"   and v :: "'a" shows "chosen u v = (SOME p. gen_iapath (verts3 (with_proj G)) u p v)"
uint8_mod_def: fixes x :: "uint8"   and y :: "uint8" shows "uint8_mod x y = snd (uint8_divmod x y)"
emeasure_empty: fixes M :: "'c measure" shows "emeasure M {} = 0"
has_domain_iff_arr: fixes f :: "'a" shows "(B.domains f \<noteq> {}) = B.arr f"
inf_hf_def: fixes a :: "hf"   and b :: "hf" shows "a \<sqinter> b = \<lbrace>x \<^bold>\<in> a. x \<^bold>\<in> b\<rbrace>"
in_Gr: fixes x :: "'d"   and y :: "'e"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "((x, y) \<in> BNF_Def.Gr A f) = (x \<in> A \<and> f x = y)"
ide_char': fixes a :: "'d" shows "Dop.ide a = (Dop.arr a \<and> (Dop.dom a = a \<or> Dop.cod a = a))"
well_valued0_rchild_if_well_valued0: fixes as :: "'b list"   and t :: "'b tree" assumes "well_valued0 as t" shows "well_valued0 as (rchild t)"
of_rat_less: fixes r :: "rat"   and s :: "rat" shows "(of_rat r < of_rat s) = (r < s)"
eventually_ex: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'b filter" shows "(\<forall>\<^sub>F x in F. \<exists>y. P x y) = (\<exists>Y. \<forall>\<^sub>F x in F. P x (Y x))"
finite_fset: fixes S :: "'x fset" shows "finite (fset S)"
anycast_simps_7: fixes P :: "'a \<Rightarrow> bool"   and m :: "'a" shows "anycast P (receive m) = True"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "L.isomorphic a a' = (\<exists>f. L.in_hom f a a' \<and> L.iso f)"
nat: fixes x :: "'a"   and n :: "nat" shows "f (x + of_nat n * a) = f x"
\<Delta>_simps_1: shows "seq (\<chi> \<star> s\<^sub>0) \<sigma>"
norm_minus_cancel: 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 x :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (uminus x) = norm x"
ts_tuple_rel_set_filter: fixes x :: "nat \<times> 'a option list"   and P :: "nat \<times> 'a option list set \<Rightarrow> bool"   and xs :: "(nat \<times> 'a option list set) list" assumes "x \<in> ts_tuple_rel (set (filter P xs))" shows "x \<in> ts_tuple_rel (set xs)"
r_univuniv2_def: shows "r_univuniv2 \<equiv> Cn 3 r_phi [Cn 3 (r_universal 2) [recf.Id 3 0, recf.Id 3 0, recf.Id 3 1], recf.Id 3 2]"
id1: shows "rat_of_int (coeff (pCons b c2) (degree q) div lead_coeff q) = rat_of_int (coeff (pCons b c2) (degree q)) / rat_of_int (lead_coeff q)"
has_unique_favorites_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" shows "has_unique_favorites R = (\<forall>i. favorites R i = {} \<or> is_singleton (favorites R i))"
abs_summable_on_normI: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "f abs_summable_on A" shows "(\<lambda>x. norm (f x)) abs_summable_on A"