hcju/isabelleps huggingface dataset:
assms_6: fixes p :: "(_) element_ptr"   and shadow_root_option :: "(_) shadow_root_ptr option"   and shadow_root_option' :: "(_) shadow_root_ptr option" assumes "h \<turnstile> get_shadow_root p \<rightarrow>\<^sub>r shadow_root_option"   and "h' \<turnstile> get_shadow_root p \<rightarrow>\<^sub>r shadow_root_option'" shows "if shadow_root_option = None then shadow_root_option' = None else shadow_root_option' = None \<or> shadow_root_option' = shadow_root_option"
that: shows "constant_term (snd x) \<noteq> (0::'a)"
prj_simps_5: shows "C.dom gf.prj\<^sub>0 = g.leg0 \<down>\<down> f.leg1"
ideal_generated_repeated: fixes a :: "'a" shows "ideal_generated {a, a} = ideal_generated {a}"
merges_preserves_type_lemma: fixes n :: "nat"   and ps :: "(nat \<times> 'a) list" shows "\<forall>xs. xs \<in> list n A \<longrightarrow> (\<forall>(p, x)\<in>set ps. p < n \<and> x \<in> A) \<longrightarrow> merges f ps xs \<in> list n A"
prems: shows "\<forall>x\<in>fv (Var x). (\<exists>f. \<theta> x = Fun f []) \<or> (\<exists>y. \<theta> x = Var y)"
path_empty_simp: fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and v :: "'a"   and v' :: "'a" shows "Pre_Port_Graph.path edges v v' [] = (v' = v)"
can_sync_devreserp_d: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and frs :: "'addr frame list"   and h' :: "'heap"   and L :: "('addr + 'thread_id + 'thread_id) set"   and h :: "'heap" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "execd_mthr.can_sync P t (xcp, frs) h' L"   and "\<Phi> \<turnstile> t:(xcp, h, frs) \<surd>"   and "h \<unlhd> h'"   and "hconf h'" shows "\<exists>L'\<subseteq>L. execd_mthr.can_sync P t (xcp, frs) h L'"
Ssegment_inc_if: fixes D :: "'c Order"   and b :: "'c"   and a :: "'c" assumes "Order D"   and "b \<in> carrier D"   and "a \<in> Ssegment D b" shows "a \<prec>\<^bsub>D\<^esub> b"
RestrictedP_graph_aux_def: shows "RestrictedP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>RestrictedP_aux x y f k g. x1 = (f, k, g) \<and> x2 = g SUBS f AND SyntaxN.All x (SyntaxN.All y (HPair (Var x) (Var y) IN g IFF Var x IN k AND HPair (Var x) (Var y) IN f)) \<and> atom x \<sharp> (y, f, k, g) \<and> atom y \<sharp> (f, k, g))"
hcomplex_equality: fixes z :: "complex star"   and w :: "complex star" assumes "hRe z = hRe w"   and "hIm z = hIm w" shows "z = w"
inf_selective: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linear_lattice inf less_eq less sup" shows "inf x y = x \<or> inf x y = y"
one_minus_const_fps_X_neg_power': fixes n :: "nat"   and c :: "'a" assumes "0 < n" shows "inverse ((1 - fps_const c * fps_X) ^ n) = Abs_fps (\<lambda>k. of_nat (n + k - 1 choose k) * c ^ k)"
multpw_converse: fixes ns :: "('a \<times> 'a) set" shows "multpw (ns\<inverse>) = (multpw ns)\<inverse>"
add_right_mono: fixes a :: "'i multiset"   and b :: "'i multiset"   and c :: "'i multiset" assumes "a \<subseteq># b" shows "a + c \<subseteq># b + c"
6: shows "g (m, l) * f (l, l)\<^sup>\<star> \<le> g (m, l)"
inf_left_commute: fixes x :: "'f multiset"   and y :: "'f multiset"   and z :: "'f multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
rel_distinct_10: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y6 :: "'b rexp" shows "\<not> rel_rexp R (Star y6) Zero"
smallo_abs_powr_iff: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "0 < p" shows "((\<lambda>x. \<bar>f x\<bar> powr p) \<in> o[F](\<lambda>x. \<bar>g x\<bar> powr p)) = (f \<in> o[F](g))"
real_mult_eq_0_iff: fixes a :: "'a"   and b :: "'a" assumes "is_real a"   and "is_real b" shows "(a * b = (0::'a)) = (a = (0::'a) \<or> b = (0::'a))"
mult_commute: fixes a :: "'a"   and b :: "'a" shows "a \<oplus> b = b \<oplus> a"
connected_in_V_1: fixes v :: "Vertices"   and w :: "Vertices" assumes "connected v w" shows "v \<in> V\<^bsub>T2\<^esub>"
Bound: fixes a :: "name"   and y :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a<\<nu>y> \<prec> Q'"   and "y \<sharp> P"   and "y \<sharp> Q"   and "y \<sharp> C" shows "\<exists>P'. P \<longmapsto> a<\<nu>y> \<prec> P' \<and> (P', Q') \<in> Rel"
SUP_union: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and M :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (M ` (A \<union> B)) = sup (Sup (M ` A)) (Sup (M ` B))"
igSubstIGAbs_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSubstIGAbs MOD \<equiv> \<forall>ys y Y xs x s X. isInBar (xs, s) \<and> igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
cod_comp: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d"   and f :: "'d" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.cod C (C g f) = partial_magma.cod C g"
mem_uminus: fixes x :: "'a \<Rightarrow> real" assumes "x \<in> diff_fun_space" shows "- x \<in> diff_fun_space"
nop_n: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop A n f" shows "n \<in>\<^sub>\<circ> \<omega>"
lossless_spmf_conv_spmf_of_pmf: fixes p :: "'c spmf" shows "lossless_spmf p = (\<exists>p'. p = spmf_of_pmf p')"
check_precond_safe_distance: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "check_precond s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o = safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"
initials_in_vertices: fixes G :: "diagram" shows "initials G |\<subseteq>| G^V"
weighted_RP_fair: shows "fair_state_seq (lmap state_of_wstate Sts)"
rel_Grp: fixes A1 :: "'d set"   and f1 :: "'d \<Rightarrow> 'g"   and A2 :: "'e set"   and f2 :: "'e \<Rightarrow> 'h" shows "rel_gpv (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. results'_gpv x \<subseteq> A1 \<and> outs'_gpv x \<subseteq> A2} (map_gpv f1 f2)"
modifies_split: fixes vs :: "char list set"   and l :: "char list \<Rightarrow> int \<Rightarrow> int"   and g :: "char list \<Rightarrow> int \<Rightarrow> int"   and l' :: "char list \<Rightarrow> int \<Rightarrow> int"   and g' :: "char list \<Rightarrow> int \<Rightarrow> int" shows "modifies vs <l|g> <l'|g'> = (modifies (Collect is_global \<union> vs) l l' \<and> modifies ({a. is_local a} \<union> vs) g g')"
butlast_append: fixes xs :: "'d list"   and ys :: "'d list" shows "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
mat_mult: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'a list list"   and nc :: "nat"   and m2 :: "'a list list"   and ze :: "'a"   and pl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and ti :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "mat nr n m1"   and "mat n nc m2" shows "mat nr nc (mat_multI ze pl ti nr m1 m2)"
preserves_arr: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "arr (\<F>C.D f)"
DERIV_NS_iff: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a" shows "(f has_field_derivative D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'a\<rightarrow>\<^sub>N\<^sub>S D"
DisjointPaths_extend: fixes P :: "'a list" assumes "v0 \<leadsto>P\<leadsto>\<^bsub>G\<^esub> v1"   and "\<And>xs v. \<lbrakk>xs \<in> paths; xs \<noteq> P; v \<in> set xs; v \<in> set P\<rbrakk> \<Longrightarrow> v = v0 \<or> v = v1" shows "DisjointPaths G v0 v1 (insert P paths)"
dim_col_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'c" shows "dim_col (mat nr nc f) = nc"
F_det: fixes P :: "'b process"   and Q :: "'b process" shows "F (P [+] Q) = {(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}"
finite_half_joinpaths_inter: fixes l1 :: "real \<Rightarrow> 'a"   and c :: "'a"   and d :: "real"   and l2 :: "real \<Rightarrow> 'a" assumes "finite {t. l1 t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"   and "finite {t. l2 t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}" shows "finite {t. (l1 +++ l2) t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
ssum_def: shows "ssum = (\<lambda>s. sum (jump f) ({x. jump f x \<noteq> 0 \<and> a < x \<and> x < c} \<inter> s))"
box_stp_actD: fixes P :: "state \<Rightarrow> bool"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<box>$P" shows "sigma \<Turnstile> \<box>P"
contour_integral_subpath_combine: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and u :: "real"   and v :: "real"   and w :: "real" assumes "f contour_integrable_on g"   and "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "w \<in> {0..1}" shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = contour_integral (subpath u w g) f"
is_Var_def: fixes trm :: "trm" shows "is_Var trm = (\<exists>x1. trm = Var x1)"
compatAbsSubst_def: fixes xs :: "'varSort"   and s :: "'b"   and f :: "'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c" shows "compatAbsSubst xs s f \<equiv> \<forall>x X x' X'. (\<forall>Y. wls (asSort xs) Y \<longrightarrow> X #[Y / x]_xs = X' #[Y / x']_xs) \<longrightarrow> f x X = f x' X'"
subset: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set" assumes "semilattice_set f"   and "finite A"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "f (semilattice_set.F f B) (semilattice_set.F f A) = semilattice_set.F f A"
length_compl_induct: fixes P :: "'f list \<Rightarrow> bool"   and l :: "'f list" assumes "P []"   and "\<And>e l. (\<And>ll. length ll \<le> length l \<Longrightarrow> P ll) \<Longrightarrow> P (e # l)" shows "P l"
d'_neg: shows "\<forall>y>d'. y < c \<longrightarrow> poly p y < 0"
norm_blinfun_of_matrix: fixes a :: "'e \<Rightarrow> 'd \<Rightarrow> real" shows "norm (blinfun_of_matrix a) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>a i j\<bar>)"
ipurge_ref_aux_def: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref_aux I D U xs X \<equiv> {x \<in> X. \<forall>v\<in>sinks_aux I D U xs. (v, D x) \<notin> I}"
reindex_bij_witness: fixes S :: "'b set"   and i :: "'c \<Rightarrow> 'b"   and j :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"   and "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "sum g S = sum h T"
all_le_start_of: fixes ap :: "abc_inst list" shows "list_all (\<lambda>(acn, s). s \<le> start_of (layout_of ap) (length ap)) (tm_of ap)"
to_single_list: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less" shows "x # xs = linorder.single_list x @ xs"
fmapU_cast_DEFL: shows "fmapU\<cdot>(cast\<cdot>DEFL('b)) = PRJ(udom\<cdot>'a) oo cast\<cdot>DEFL('b\<cdot>'a) oo EMB(udom\<cdot>'a)"
degree_of_sum_diff_degree: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < deg R p" shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) = deg R p"
cone_elem_sum: fixes Vs :: "'a vec set"   and x :: "'a vec"   and y :: "'a vec" assumes "Vs \<subseteq> carrier_vec n"   and "x \<in> cone Vs"   and "y \<in> cone Vs" shows "x + y \<in> cone Vs"
set_trie_isin: fixes t :: "trie"   and xs :: "bool list" shows "isin t xs = (xs \<in> set_trie t)"
sel_1: fixes x1 :: "'b"   and x2 :: "'b tree"   and x3 :: "'b tree" shows "root (Node x1 x2 x3) = x1"
757: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> x \<sqinter> (- y \<sqinter> - z) = x"
pp_increasing: fixes x :: "'c" shows "- - x \<le> x"
exhaust: fixes y :: "v" obtains x1 :: "i32" where "y = ConstInt32 x1"   | x2 :: "i64" where "y = ConstInt64 x2"   | x3 :: "f32" where "y = ConstFloat32 x3"   | x4 :: "f64" where "y = ConstFloat64 x4"
inj_vreal_of_real: shows "inj vreal_of_real"
surj_def: fixes f :: "'c \<Rightarrow> 'b" shows "surj f = (\<forall>y. \<exists>x. y = f x)"
prems_4: shows "x \<in> lvars (\<T> (l ! i))"
dim_gen_is_basis: fixes A :: "'c set" assumes "finite A"   and "A \<subseteq> carrier V"   and "gen_set A"   and "card A \<le> local.dim" shows "basis A"
open_contains_cbox: fixes A :: "'b set"   and x :: "'b" assumes "open A"   and "x \<in> A" obtains a :: "'b" and b :: "'b" where "cbox a b \<subseteq> A" and "x \<in> box a b" and "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i"
inf_sup_aci_5: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> y = y \<oplus> x"
conc_def: fixes A :: "'b list set"   and B :: "'b list set" shows "A \<cdot> B = {xs @ ys |xs ys. xs \<in> A \<and> ys \<in> B}"
r_length_def: shows "r_length \<equiv> Cn 1 r_ifz [recf.Id 1 0, Z, Cn 1 S [Cn 1 r_pdec1 [r_dec]]]"
left: assumes "height r \<le> height l" shows "fib (height \<langle>l, (a, h), r\<rangle> + 2) \<le> size1 \<langle>l, (a, h), r\<rangle>"
k_2: fixes i :: "nat"   and G1 :: "'a poly"   and G2 :: "'a poly" assumes "i < ??.Subresultant.k G1 G2" shows "??.Subresultant.F G1 G2 (Suc i) \<noteq> 0"
interPrefix: fixes s :: "'c event list"   and t :: "'c event list"   and A :: "'c event set"   and u :: "'c event list"   and t1 :: "'c event list" shows "\<exists>u1 s1. s setinterleaves ((t, u), A) \<and> t1 < t \<longrightarrow> u1 \<le> u \<and> s1 < s \<and> s1 setinterleaves ((t1, u1), A)"
SumA_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" shows "A B C D E F SumA G H I \<equiv> \<exists>J. C B J CongA D E F \<and> \<not> B C OS A J \<and> Coplanar A B C J \<and> A B J CongA G H I"
hyps_3: fixes s :: "char list \<Rightarrow> int" shows "Q' s \<le> Q s"
ring_simprules_7: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"
rel_ar_def: fixes R :: "('b \<times> 'b) set" shows "rel_ar R = {(y, y) |y. \<nexists>x. (x, y) \<in> R}"
elims_2: fixes x :: "('a \<times> ('b \<Rightarrow> 'c)) list"   and xa :: "'a \<Rightarrow> 'c"   and xb :: "'b" assumes "postList x xa xb" obtains l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = []" and "xa = la__" and "xb = sa__"   | xc :: "'a" and e :: "'b \<Rightarrow> 'c" and xs :: "('a \<times> ('b \<Rightarrow> 'c)) list" and l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = (xc, e) # xs" and "xa = l" and "xb = s" and "l xc = e s \<and> postList xs l s"
analz_Un_implInsecSet: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> payload"   and "Enc_keys_clean H" shows "analz ({implInsec A B M |A B M. M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
natural_isomorphism_axioms: shows "natural_isomorphism CC.comp (\<cdot>) T ToS.map \<sigma>.map"
zero_less_oFix_eq: fixes F :: "ordinal \<Rightarrow> ordinal" shows "(0 < oFix F 0) = (0 < F 0)"
suffixI: fixes ys :: "'c list"   and zs :: "'c list"   and xs :: "'c list" assumes "ys = zs @ xs" shows "suffix xs ys"
card_inj_on_le: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "card A \<le> card B"
run_state_bind: fixes x :: "('d, 'e) state"   and f :: "'e \<Rightarrow> ('d, 'c) state"   and s :: "'d" shows "run_state (x \<bind> f) s = (case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "plus_hom hom" shows "semigroup_add_hom hom"
abs_eq: fixes xa :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'c option" shows "Mapping.map_values xa (Mapping x) = Mapping (\<lambda>xb. map_option (xa xb) (x xb))"
inject_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Var x1 = Var y1) = (x1 = y1)"
simps_1: fixes r :: "real" shows "- ereal r = ereal (- r)"
act_emp: fixes x :: "'a" shows "\<alpha> x (\<Squnion>{}) = \<bottom>"
map_cong0: fixes x :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>z. s2p (range x) z \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> x"
irrefl: fixes a :: "'b list" shows "\<not> strict_suffix a a"
special_points_on_y_axis'_3: shows "on_circline y_axis \<infinity>\<^sub>h"
arith_sum_t2k: fixes t :: "nat"   and k :: "nat" shows "t + 2 + k = Suc t + Suc k"
of_complex_0: shows "of_complex 0 = (0::'b)"
perm_eval_2: fixes x :: "'a"   and xs :: "'a list"   and l :: "'a list" shows "(x # xs <~~> l) = (case delete_first (\<lambda>e. e = x) l of None \<Rightarrow> False | Some l' \<Rightarrow> xs <~~> l')"
one_: fixes i :: "'a" assumes "i \<in> I" shows "\<one>\<^bsub>G i\<^esub> = \<one>\<^bsub>DirProds G I\<^esub> i"
set_factorization_lattice_in_carrier: fixes u :: "'a poly"   and d :: "nat"   and m :: "'a" shows "set (factorization_lattice u d m) \<subseteq> carrier_vec (degree u + d)"
ISpec_stutinv: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "STUTINV ISpec ic q oc"
liftWT_bind: fixes m :: "'h\<cdot>'f"   and k :: "'h \<rightarrow> 'e\<cdot>'f" shows "liftWT\<cdot>(m \<bind> k) = bindWT\<cdot>(liftWT\<cdot>m)\<cdot>(liftWT oo k)"
simps_5: fixes v :: "t list" shows "produce (Type v) Bot = Bot"
Gramian_determinant_def: fixes G :: "'a vec list"   and k :: "nat" shows "gso.Gramian_determinant G k = det (gso.Gramian_matrix G k)"
ennreal_tendsto_cmult: fixes c :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and x :: "ennreal"   and F :: "'c filter" assumes "c < top"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * x) F"
app_beta_Betas: fixes s :: "Lambda_Z.term"   and s' :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and t' :: "Lambda_Z.term" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'"   and "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "app_beta s t \<rightarrow>\<^sub>\<beta>\<^sup>* app_beta s' t'"
C: shows "\<langle>n, Pwd n\<rangle> \<in> spied s \<longrightarrow> (\<exists>SK. SesKey SK \<in> spied s \<and> ((Owner n, Crypt (SesK SK) (Pwd n)) \<in> s \<or> (Asset n, Crypt (SesK SK) (Num 0)) \<in> s))"
sgn_1: fixes x :: "int poly \<times> rat \<times> rat" assumes "invariant_1 x" shows "real_of_rat (sgn_1 x) = sgn (real_of_1 x)"
diagonalization: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {xx}" shows "\<exists>\<psi>. \<psi> \<in> fmla \<and> Fvars \<psi> = {} \<and> bprv (eqv \<psi> (subst \<phi> \<langle>\<psi>\<rangle> xx))"
left_connected: fixes \<nu> :: "'a"   and \<nu>' :: "'a"   and \<mu> :: "'a" assumes "seq \<nu> \<nu>'" shows "(\<nu> \<star> \<mu> \<noteq> null) = (\<nu>' \<star> \<mu> \<noteq> null)"
valid_var_NormalI: fixes A :: "'b triple set"   and G :: "prog"   and t :: "Term.var"   and P :: "'b assn"   and Q :: "'b assn" assumes "\<And>n s0 L accC T C vf s1 Y Z. \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G, L); normal s0; \<lparr>prg = G, cls = accC, lcl = L\<rparr>,empty_dt\<Turnstile>t\<Colon>=T; \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>In2 t\<guillemotright> C; G\<turnstile>s0 \<midarrow>t=\<succ>vf\<midarrow>n\<rightarrow> s1; Normal P Y s0 Z\<rbrakk> \<Longrightarrow> Q (In2 vf) s1 Z \<and> s1\<Colon>\<preceq>(G, L)" shows "G,A|\<Turnstile>\<Colon>{{Normal P} t=\<succ> {Q}}"
selsort_singleton_iff: fixes xa :: "real \<times> real"   and xs :: "(real \<times> real) list"   and x :: "real \<times> real" shows "(ccw.selsort xa xs = [x]) = (xs = [x])"
list_set_rel_finiteD: fixes xs :: "'c list"   and X :: "'d set"   and A :: "('c \<times> 'd) set" assumes "(xs, X) \<in> \<langle>A\<rangle>list_set_rel" shows "finite X"
Inl_dec_interp_length: fixes FO :: "nat set"   and w :: "('a \<times> bool list) list"   and p :: "nat"   and n :: "nat" assumes "\<forall>i\<in>FO. \<exists>!p. p < length w \<and> snd (w ! p) ! i"   and "Inl p \<in> set (dec_interp n FO w)" shows "p < length w"
abs_fresh_1: fixes b :: "name"   and a :: "name"   and x :: "'b" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
ptrm_size_prm: fixes X :: "'b ptrm"   and \<pi> :: "'b prm" shows "size X = size (\<pi> \<bullet> X)"
span_explicit: fixes b :: "'a set" assumes "b \<subseteq> UNIV" shows "span b = {x. (\<exists>t\<subseteq>UNIV. (\<exists>r. x = (\<Sum>a\<in>t. r a *\<^sub>R a)) \<and> finite t \<and> t \<subseteq> b) \<and> x \<in> UNIV}"
Red_F_\<G>_empty_q_def: fixes q :: "'q"   and N :: "'f set" shows "Red_F_\<G>_empty_q q N = {C. \<forall>D\<in>\<G>_F_q q C. D \<in> Red_F_q q (\<G>_Fset_q q N)}"
mult_left_inj: fixes n :: "nat" assumes "0 < n" shows "inj ((*) n)"
element_ord_generates_cyclic: fixes a :: "'a" assumes "a \<in> carrier G"   and "ord a = order G" shows "Finite_And_Cyclic_Groups.cyclic_group G a"
101: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> x \<sqinter> - - z) = y \<squnion> x"
bind_gpv'_cong: fixes gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv"   and f :: "'d option \<Rightarrow> ('g, 'e, 'f) gpv"   and f' :: "'d option \<Rightarrow> ('g, 'e, 'f) gpv" assumes "gpv = gpv'"   and "\<And>x. x \<in> Some ` results'_gpv gpv' \<or> \<not> colossless_gpv \<I>_full gpv \<and> x = None \<Longrightarrow> f x = f' x" shows "catch_gpv gpv \<bind> f = catch_gpv gpv' \<bind> f'"
cmpghom_def: fixes F :: "('e, 'g) Group_scheme"   and g :: "'h \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'h" shows "g \<circ>\<^bsub>F\<^esub> f = compose (carrier F) g f"
expr_sem_rf_sound: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and V :: "nat set"   and \<sigma> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile> e : t"   and "randomfree e"   and "free_vars e \<subseteq> V"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "return_val (expr_sem_rf \<sigma> e) = expr_sem \<sigma> e"
rot_eq_0_iff: fixes x :: "'a" shows "(rot x = (0::'a)) = (x = (0::'a))"
assms_2: shows "\<forall>m\<in>M. P m"
map_cong_pred: fixes x :: "'b list"   and ya :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "list_all (\<lambda>z. f z = g z) ya" shows "map f x = map g ya"
map_upd_upds_conv_if: fixes f :: "'c \<Rightarrow> 'd option"   and x :: "'c"   and y :: "'d"   and xs :: "'c list"   and ys :: "'d list" shows "f(x \<mapsto> y, xs [\<mapsto>] ys) = (if x \<in> set (take (length ys) xs) then f(xs [\<mapsto>] ys) else f(xs [\<mapsto>] ys, x \<mapsto> y))"
karatsuba_mult_poly: fixes f :: "'b poly"   and g :: "'b poly" shows "karatsuba_mult_poly f g = f * g"
det_mk_poly_sub: fixes A :: "'b poly mat"   and n :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat n n"   and "i < n" shows "det (mk_poly_sub A (n - 1) i) = det A"
c1: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> x = x"
sq_norm_poly_def: fixes p :: "'a poly" shows "\<parallel>p\<parallel>\<^sup>2 \<equiv> sum_list (map sq_norm (coeffs p))"