hcju/isabelleps huggingface dataset:
finfun_Diag_conv_Abs_finfun: fixes f :: "('a, 'b) finfun"   and g :: "('a, 'c) finfun" shows "finfun_Diag f g = Abs_finfun (\<lambda>x. (finfun_apply f x, finfun_apply g x))"
nth_n: shows "lnth (lconcat (LCons xs' xss'')) n = lnth (lnth (LCons xs' xss'') m) n'"
independent_some_basis: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "vector_space scale" shows "\<not> module.dependent scale (vector_space.some_basis scale V)"
Init_eq: fixes init :: "'d set"   and acts :: "('d \<times> 'd) set set"   and allowed :: "('d \<times> 'd) set set" shows "Init (mk_program (init, acts, allowed)) = init"
dist_triangle_third: fixes x1 :: "'b"   and x2 :: "'b"   and e :: "real"   and x3 :: "'b"   and x4 :: "'b" assumes "dist x1 x2 < e / 3"   and "dist x2 x3 < e / 3"   and "dist x3 x4 < e / 3" shows "dist x1 x4 < e"
alw_nxt_holds_cong: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and n :: "nat"   and Q :: "'b \<Rightarrow> bool" assumes "alw (holds P) xs" shows "(nxt ^^ n) (holds (\<lambda>x. P x \<and> Q x)) xs = (nxt ^^ n) (holds Q) xs"
HNatInfinite_diff: fixes x :: "nat star"   and y :: "nat star" assumes "x \<in> HNatInfinite"   and "y \<in> \<nat>" shows "x - y \<in> HNatInfinite"
emeasure_pmf_of_set: fixes S :: "'c set"   and A :: "'c set" assumes "S \<noteq> {}"   and "finite S" shows "emeasure (measure_pmf (pmf_of_set S)) A = ennreal (real (card (S \<inter> A)) / real (card S))"
IH: fixes U :: "variable set" assumes "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg" shows "U \<union> BVG (the (snd (usubstappp \<sigma> U \<alpha>))) \<subseteq> fst (usubstappp \<sigma> U \<alpha>)"
is_denormal_Float_def: fixes x :: "('a, 'b) IEEE.float itself"   and f :: "Float.float" shows "is_denormal_Float x f = (mantissa f \<noteq> 0 \<and> bitlen \<bar>mantissa f\<bar> \<le> 1 - Float.exponent f - int (bias x) \<and> 1 - 2 ^ (LENGTH('a) - 1) - int LENGTH('b) < Float.exponent f)"
Limit_vproduct_in_VsetI: fixes \<alpha> :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "Limit \<alpha>"   and "I \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> A i \<in>\<^sub>\<circ> Vset \<alpha>"   and "vfinite I" shows "vproduct I A \<in>\<^sub>\<circ> Vset \<alpha>"
tllist_all2_TCons: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and x21 :: "'g"   and x22 :: "('g, 'i) tllist"   and y21 :: "'h"   and y22 :: "('h, 'j) tllist" shows "tllist_all2 R1 R2 (TCons x21 x22) (TCons y21 y22) = (R1 x21 y21 \<and> tllist_all2 R1 R2 x22 y22)"
m_s2_rep: fixes X :: "'a set"   and S1 :: "'a \<Rightarrow> 'av"   and S2 :: "'a \<Rightarrow> 'av" assumes "finite X"   and "S1 = S2 on - X"   and "\<forall>x. S1 x \<le> S2 x"   and "S1 \<noteq> S2" shows "(\<Sum>x\<in>X. m (S2 x)) < (\<Sum>x\<in>X. m (S1 x))"
\<L>_def: shows "\<L> = {v |v. is_word v \<and> is_derivation v}"
rotate_rotate: fixes m :: "nat"   and n :: "nat"   and xs :: "'c list" shows "rotate m (rotate n xs) = rotate (m + n) xs"
H_inv: shows "H * H = 1\<^sub>m 2"
hyps: shows "\<forall>m<card A. \<forall>x. m = card x \<longrightarrow> finite x \<longrightarrow> x \<subseteq> carrier G \<longrightarrow> (\<exists>gs. set gs \<subseteq> generate G x \<and> cycle gs \<and> is_idirprod (generate G x) (\<lambda>g. generate G {g}) (set gs) \<and> successively (dvd) (map ord gs) \<and> card (set gs) \<le> card x)"
mult_indicator_subset: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<subseteq> B" shows "indicator A x * indicator B x = indicator A x"
cB: shows "char_poly B = (\<Prod>a\<leftarrow>bs. [:- a, 1::'a:])"
convergent_powser'_imp_convergent_powser: fixes cs :: "real msllist"   and f :: "real \<Rightarrow> real" assumes "convergent_powser' cs f" shows "convergent_powser cs"
offset_hmset_def: fixes M :: "hmultiset" shows "offset_hmset M = count (hmsetmset M) 0"
odd_bit_iff_bit_pred: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "bit a n = (bit (minus a one) n \<or> n = 0)"
pnet_net_ips_net_tree_ips: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree"   and I :: "'d node_action \<Rightarrow> bool" assumes "s \<in> reachable (pnet np p) I" shows "net_ips s = net_tree_ips p"
fT0: fixes x :: "complex" assumes "x \<in> T" shows "f x = 0"
healthy_bounded_byD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "real"   and P :: "'c \<Rightarrow> real" assumes "healthy t"   and "bounded_by b P"   and "nneg P" shows "bounded_by b (t P)"
his_act: fixes n' :: "nat"   and nid' :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "(n', nid') \<in> his t n nid" shows "\<parallel>nid'\<parallel>\<^bsub>t n'\<^esub>"
norm_le_zero_iff: 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 x \<le> 0) = (x = zero)"
wadjust_on_left_moving_singleton: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list" assumes "wadjust_erase2 m rs (c, [])"   and "c \<noteq> []" shows "wadjust_on_left_moving m rs (tl c, [hd c])"
mult_eSuc: fixes m :: "eint"   and n :: "eint" shows "(m + 1) * n = m * n + n"
pullback_commutes': fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "f \<cdot> \<p>\<^sub>1[f, g] = g \<cdot> \<p>\<^sub>0[f, g]"
times_MP_Rel: shows "(MP_Rel ===> MP_Rel ===> MP_Rel) (*) (*)"
hyps: fixes w :: "'a list" assumes "\<^bold>|w\<^bold>| < \<^bold>|wa__\<^bold>|"   and "bordered w" shows "\<exists>x. x \<le>b w \<and> \<not> bordered x"
Red_F_of_Red_F_subset: fixes N' :: "'f set"   and N :: "'f set" assumes "N' \<subseteq> Red_F N" shows "Red_F N \<subseteq> Red_F (N - N')"
pr_comp2_1': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat" assumes "pr_conv_2_to_1 f \<in> PrimRec1'"   and "g \<in> PrimRec1'"   and "h \<in> PrimRec1'" shows "(\<lambda>x. f (g x) (h x)) \<in> PrimRec1'"
hom_pow: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c"   and n :: "nat" assumes "ring_hom_cring R S h"   and "x \<in> carrier R" shows "h (x [^]\<^bsub>R\<^esub> n) = h x [^]\<^bsub>S\<^esub> n"
comp_cod_arr: fixes f :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VoV.arr f"   and "VoV.cod f = b" shows "VoV.comp b f = f"
lookup_map_entry: fixes x :: "'d"   and f :: "'c \<Rightarrow> 'c"   and m :: "('d, 'c) mapping" shows "Mapping.lookup (Mapping.map_entry x f m) x = map_option f (Mapping.lookup m x)"
assms_3: fixes i :: "'a" assumes "i \<in> Basis" shows "x \<bullet> i \<in> X i"
lowner_le_refl: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "A \<le>\<^sub>L A"
simps: fixes a :: "'b set"   and f :: "'b perm" shows "(a \<in> orbits f) = (\<exists>aa. a = orbit f aa \<and> aa \<in> affected f)"
constructive_security_obsf'D: fixes \<eta> :: "nat" shows "constructive_security_obsf (real \<eta>) (middle \<eta>) (sim_outer \<eta>) (\<I>_real \<eta>) (\<I>_middle \<eta>) (\<I>_common \<eta>) (\<A> \<eta>) (advantage (\<A> \<eta>) (obsf_resource (sim_outer \<eta> |\<^sub>= 1\<^sub>C \<rhd> middle \<eta>)) (obsf_resource (real \<eta>)))"
g1: shows "filter (\<lambda>y. f x = f y) xs = map fst (filter (\<lambda>y. f x = snd y) (map (\<lambda>x. (x, f x)) xs))"
poly_nth_root_def: fixes p :: "'b poly" shows "local.poly_nth_root p = p \<circ>\<^sub>p monom (1::'b) n"
tm_shift_fetch: fixes A :: "(action \<times> nat) list"   and s :: "nat"   and b :: "cell"   and ac :: "action"   and ns :: "nat"   and off :: "nat" assumes "fetch A s b = (ac, ns)"   and "ns \<noteq> 0" shows "fetch (shift A off) s b = (ac, ns + off)"
inj_monoid_mult_hom_axioms: shows "inj_monoid_mult_hom (inv (inv hom))"
f_Exec_Stream_not_empty_conv: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "(f_Exec_Comp_Stream trans_fun xs c \<noteq> []) = (xs \<noteq> [])"
rmMatchFalse_helper: fixes m :: "'a match_expr"   and a :: "action"   and rsa :: "'a rule list" assumes "m \<noteq> MatchNot MatchAny" shows "rmMatchFalse (Rule m a # rsa) = Rule m a # rmMatchFalse rsa"
type_less_OclVoid_x_intro: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<noteq> OclVoid" shows "OclVoid < \<tau>"
absolutely_integrable_on_def: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "(f absolutely_integrable_on S) = (f integrable_on S \<and> (\<lambda>x. norm (f x)) integrable_on S)"
join1_wf_tuple: fixes v1 :: "'b option list"   and v2 :: "'b option list"   and v :: "'b option list"   and n :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "join1 (v1, v2) = Some v"   and "wf_tuple n A v1"   and "wf_tuple n B v2" shows "wf_tuple n (A \<union> B) v"
induct: fixes x1 :: "wall"   and x2 :: "wall"   and P :: "wall \<Rightarrow> wall \<Rightarrow> bool" assumes "x1 ~f x2"   and "\<And>a. P a a"   and "\<And>a b. framed_linkrel a b \<Longrightarrow> P a b"   and "\<And>a. domain_wall a = 0 \<and> is_tangle_diagram a \<Longrightarrow> P a (basic [] \<circ> a)"   and "\<And>a. codomain_wall a = 0 \<and> is_tangle_diagram a \<Longrightarrow> P a (a \<circ> basic [])"   and "\<And>B D A C. (B ~f D \<and> P B D) \<and> (A ~f C \<and> P A C) \<and> is_tangle_diagram A \<and> is_tangle_diagram B \<and> is_tangle_diagram C \<and> is_tangle_diagram D \<and> domain_wall B = codomain_wall A \<and> domain_wall D = codomain_wall C \<Longrightarrow> P (A \<circ> B) (C \<circ> D)"   and "\<And>A B C. \<lbrakk>A ~f B; P A B; B ~f C; P B C\<rbrakk> \<Longrightarrow> P A C"   and "\<And>A B. \<lbrakk>A ~f B; P A B\<rbrakk> \<Longrightarrow> P B A"   and "\<And>B D A C. (B ~f D \<and> P B D) \<and> (A ~f C \<and> P A C) \<and> is_tangle_diagram A \<and> is_tangle_diagram B \<and> is_tangle_diagram C \<and> is_tangle_diagram D \<Longrightarrow> P (A \<otimes> B) (C \<otimes> D)" shows "P x1 x2"
index_last: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "distinct xs" shows "index xs (last xs) = length xs - 1"
zip_with_map_left: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and xs :: "'e list"   and ys :: "'d list" shows "zip_with f (map g xs) ys = zip_with (\<lambda>x. f (g x)) xs ys"
sub1: fixes A :: "'a"   and k :: "'a"   and a :: "'a"   and b :: "'a" assumes "A \<equiv> k + a" shows "A - b \<equiv> k + (a - b)"
nat_pow_Suc: fixes x :: "'a mod_ring poly"   and n :: "nat" shows "x [^]\<^bsub>R\<^esub> Suc n = x [^]\<^bsub>R\<^esub> n \<otimes>\<^bsub>R\<^esub> x"
simps_2: fixes n :: "nat \<Rightarrow> nat"   and k :: "nat" shows "inf_concat n (Suc k) = (let (i, j) = inf_concat n k in if Suc j < n i then (i, Suc j) else (LEAST i'. i < i' \<and> 0 < n i', 0))"
distinct_23: fixes x3 :: "bool"   and x61 :: "Unary_PCF.type"   and x62 :: "expr" shows "x3 \<noteq> \<Lambda>\<langle>x61\<rangle> x62"
full_chain_LNil: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "\<not> full_chain R LNil"
sum_constant_scale: fixes y :: "'a"   and A :: "'b set" shows "(\<Sum>x\<in>A. y) = of_nat (card A) * y"
empty_empty: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and m :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ [Rule m Empty] @ rs\<^sub>2, s\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, s\<rangle> \<Rightarrow> t"
S2: shows "pr_conv_1_to_2 (nat_to_pr n) \<in> PrimRec2"
s_2: fixes x :: "'e"   and y :: "'a" assumes "x \<in> space N"   and "y \<in> space M" shows "(\<lambda>i. s i (x, y)) \<longlonglongrightarrow> f x y"
ir_Assign: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and e :: "aexp"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare (\<lambda>t t'. \<exists>v. P (t(x := v)) t' \<and> t x = aval e (t(x := v))) SKIP c' Q" shows "ir_hoare P (x ::= e) c' Q"
rxs_ys_def: shows "rxs_ys = map (\<lambda>(x, y). (rat_of_int x, rat_of_int y)) xs_ys"
SLC_get: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and l :: "'l"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Read (VE (Loc l))])"   and "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (the ((\<sigma>;;\<tau>) l))]))"   and "l \<in> dom (\<sigma>;;\<tau>)"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
f_def: shows "f = (\<lambda>x. ln (d * x + b) * (d * x + b) powr c)"
pred_ss_V: fixes u :: "nat" shows "E\<inverse> `` {u} \<subseteq> V"
subG: fixes A :: "'a set" shows "generate G (carrier G \<inter> A) \<subseteq> carrier G"
assms_2: shows "y \<in> \<langle>\<C>\<rangle>"
intros: fixes operations :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list"   and \<tau> :: "'a"   and name :: "String.literal"   and \<pi> :: "'a list"   and oper' :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option"   and oper :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" assumes "operation' operations \<tau> name \<pi> oper'"   and "oper \<noteq> oper'" shows "operation_not_unique operations \<tau> name \<pi> oper"
set_filt_contain: fixes A :: "bool stream set"   and n :: "nat"   and w :: "bool stream" assumes "A \<in> sets (nat_filtration n)"   and "w \<in> A" shows "pseudo_proj_True n -` {pseudo_proj_True n w} \<subseteq> A"
zero_index: fixes i :: "'e" shows "0 $h i = (0::'d)"
Quotient3_real: shows "Quotient3 realrel Real.Real rep_real"
convolution_delta_left: fixes b :: "'b"   and a :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "convolution (b \<delta> a) f = (\<lambda>x. b * f (- a + x))"
HId_is_gate: fixes n :: "nat" assumes "1 \<le> n" shows "gate (n + 1) (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i mod 2 = j mod 2 then (- 1) ^ i div 2 \<cdot>\<^bsub>n\<^esub>  j div 2 / sqrt 2 ^ n else 0)))"
support_on_simps_2: fixes x :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "support_on (insert x S) f = (if f x = (0::'b) then support_on S f else insert x (support_on S f))"
simps_2: fixes r :: "'b memref"   and rs :: "'b memref list" shows "read_tmps (r # rs) = (case r of Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> insert t (read_tmps rs) | _ \<Rightarrow> read_tmps rs)"
type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap" assumes "ShadowRootClass.type_wf h" shows "DocumentClass.type_wf h"
assms_3: shows "\<turnstile> \<box>F \<and> G \<longrightarrow> H"
inf_assoc: fixes x :: "'f multiset"   and y :: "'f multiset"   and z :: "'f multiset" shows "x \<inter># y \<inter># z = x \<inter># (y \<inter># z)"
span_eq_iff: fixes s :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "(phull s = s) = phull.subspace s"
op_inter_fst_ivl_coll_scaleR2_def: fixes X :: "('c \<times> 'd) set"   and Y :: "'c set" shows "op_inter_fst_ivl_coll_scaleR2 X Y = sets_of_coll X \<bind> (\<lambda>Xs. FORWEAK Xs (RETURN op_empty_coll) (\<lambda>X. op_inter_fst_ivl_scaleR2 X Y) (\<lambda>X X'. RETURN (X' \<union> X)))"
bounded_uminus: fixes X :: "'a set" shows "bounded (uminus ` X) = bounded X"
exponent_Float: fixes m :: "int"   and e :: "int" shows "Float.exponent (Float m e) = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e) - bitlen \<bar>mantissa (Float m e)\<bar>"
isNode2_def: fixes Kl :: "('d + 'e) list set"   and lab :: "('d + 'e) list \<Rightarrow> 'f + ('g, 'd, 'e) F2"   and kl :: "('d + 'e) list" shows "isNode2 Kl lab kl = (\<exists>x2. lab kl = Inr x2 \<and> F2set2 x2 = Succ1 Kl kl \<and> F2set3 x2 = Succ2 Kl kl)"
shows_list_ratfps_def: fixes ps :: "'a ratfps list" shows "shows_list ps = showsp_list shows_prec 0 ps"
ivls_of_sets_def: fixes X :: "'c set" shows "ivls_of_sets X = (sets_of_coll (X ::: clw_rel appr_rel) ::: \<langle>\<langle>appr_rel\<rangle>list_wset_rel\<rangle>nres_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN (op_empty_coll ::: clw_rel lvivl_rel)) (\<lambda>X. op_ivl_rep_of_set X \<bind> (\<lambda>(i, s). RETURN (mk_coll (op_atLeastAtMost_ivl i s ::: lvivl_rel)))) (\<lambda>a b. RETURN (a \<union> b)))"
insertion_monom: fixes f :: "nat \<Rightarrow> 'b"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and c :: "'b" shows "insertion f (MPoly_Type.monom m c) = c * (\<Prod>x. f x ^ lookup m x)"
field_potent_nonzero: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "x^\<^bsup>K n\<^esup> \<noteq> \<zero>"
diff_le: fixes y :: "real" assumes "real_of_float l \<le> y"   and "y \<le> real_of_float u" shows "real_of_float a \<le> f' y"
apfst_convE: fixes q :: "'a \<times> 'b"   and f :: "'d \<Rightarrow> 'a"   and p :: "'d \<times> 'b" assumes "q = apfst f p" obtains x :: "'d" and y :: "'b" where "p = (x, y)" and "q = (f x, y)"
cexpr_sem_cexpr_subst: fixes \<sigma> :: "nat \<Rightarrow> val"   and e :: "cexpr"   and e' :: "cexpr" shows "cexpr_sem \<sigma> (cexpr_subst 0 e e') = cexpr_sem (case_nat (cexpr_sem \<sigma> e) \<sigma>) e'"
inducts: fixes x1 :: "bool"   and x2 :: "v sem_env"   and x3 :: "'b state"   and x4 :: "top0 list"   and x5 :: "'b state \<times> (v sem_env, v) result"   and P :: "bool \<Rightarrow> v sem_env \<Rightarrow> 'b state \<Rightarrow> top0 list \<Rightarrow> 'b state \<times> (v sem_env, v) result \<Rightarrow> bool" assumes "evaluate_prog x1 x2 x3 x4 x5"   and "\<And>ck env s. P ck env s [] (s, Rval (make_sem_env nsEmpty nsEmpty))"   and "\<And>ck s1 s2 s3 env top0 tops new_env r. evaluate_top ck env s1 top0 (s2, Rval new_env) \<and> evaluate_prog ck (extend_dec_env new_env env) s2 tops (s3, r) \<and> P ck (extend_dec_env new_env env) s2 tops (s3, r) \<Longrightarrow> P ck env s1 (top0 # tops) (s3, combine_dec_result new_env r)"   and "\<And>ck s1 s2 env top0 tops err. evaluate_top ck env s1 top0 (s2, Rerr err) \<Longrightarrow> P ck env s1 (top0 # tops) (s2, Rerr err)" shows "P x1 x2 x3 x4 x5"
simps_2: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree" shows "size1 \<langle>l, x, r\<rangle> = size1 l + size1 r"
trm_simp_4: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm"   and y :: "'b"   and S :: "PreSimplyTyped.type"   and B :: "'b trm" assumes "Fn x T A = Fn y S B" shows "x = y \<and> T = S \<and> A = B \<or> x \<noteq> y \<and> T = S \<and> x \<notin> fvs B \<and> A = [x \<leftrightarrow> y] \<cdot> B"
upper_asymptotic_density_bdd_interval_2: fixes N :: "nat" shows "upper_asymptotic_density {..N} = 0"
Spy_analz_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> otway" shows "(Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
FDOM_eff_subset_FDOM_valid_states: fixes p :: "('a, 'b) fmap"   and e :: "('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and s :: "('a, 'd) fmap" assumes "(p, e) \<in> PROB"   and "s \<in> valid_states PROB" shows "fmdom' e \<subseteq> fmdom' s"
cobounded2: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) \<le> b"
of_seq_rec: fixes f :: "nat \<Rightarrow> 'a" shows "of_seq f = f 0 ## of_seq (f \<circ> Suc)"
disj: shows "pairwise (\<lambda>x y. disjnt (F x) (F y)) A"
unit_table_wf_tuple: fixes V :: "nat set"   and x :: "'b option list"   and n :: "nat" assumes "V = {}"   and "x \<in> unit_table n" shows "wf_tuple n V x"
subtype_trans_1: fixes \<Gamma> :: "binding list"   and S :: "POPLmark.type"   and Q :: "POPLmark.type"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile> S <: Q"   and "\<Gamma> \<turnstile> Q <: T" shows "\<Gamma> \<turnstile> S <: T"
CICod: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T"   and "[A,e,B]\<^bsub>T\<^esub> \<in> mor\<^bsub>CanonicalCat T\<^esub>" shows "cod\<^bsub>CanonicalCat T\<^esub> [A,e,B]\<^bsub>T\<^esub> = B"
seqsim_sym: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a" assumes "s \<approx> t" shows "t \<approx> s"
tri_betw_no_path: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and b' :: "'b"   and c' :: "'b"   and a' :: "'b" assumes "MinkowskiChain \<E> \<P> betw"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c"   and "betw a b' c"   and "betw b c' a"   and "betw c a' b" shows "\<not> (\<exists>Q\<in>\<P>. a' \<in> Q \<and> b' \<in> Q \<and> c' \<in> Q)"
fmadd_empty_1: fixes m :: "('a, 'b) fmap" shows "fmempty ++\<^sub>f m = m"
has_derivative_continuous: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'f"   and x :: "'e"   and s :: "'e set" assumes "(f has_derivative f') (at x within s)" shows "continuous (at x within s) f"
list_symp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "symp R" shows "symp (list_all2 R)"
IH: fixes \<eta>1 :: "val set list"   and \<sigma> :: "ty"   and \<eta>2 :: "val set list" shows "T \<tau> (\<eta>1 @ T \<sigma> (\<eta>1 @ \<eta>2) # \<eta>2) = T (subst (length \<eta>1) \<sigma> \<tau>) (\<eta>1 @ \<eta>2)"
one_not_zero: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
RDirProd_hom1: fixes R :: "('a, 'b) ring_scheme" shows "(\<lambda>a. (a, a)) \<in> ring_hom R (RDirProd R R)"
mkPoint_in_hom: fixes a :: "'s"   and t :: "'s" assumes "ide a"   and "t \<in> local.set a" shows "\<guillemotleft>mkPoint a t : unity \<rightarrow> a\<guillemotright>"
countable_Int2: fixes B :: "'f set"   and A :: "'f set" assumes "countable B" shows "countable (A \<inter> B)"
is_bij_def: fixes x :: "'a" shows "is_bij x \<equiv> is_map x \<and> is_inj x \<and> is_sur x"