hcju/isabelleps huggingface dataset:
sup_absorb1: fixes y :: "'a upred"   and x :: "'a upred" assumes "x \<sqsubseteq> y" shows "(x \<or> y) = x"
RU_spawn: fixes u :: "'a"   and p :: "'b"   and u' :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and v :: "'a"   and M :: "'d set"   and P :: "'b multiset"   and q :: "'b"   and Ml :: "'d set"   and Me :: "'d set"   and h :: "'d \<Rightarrow> 'd set"   and U :: "'a set" assumes "(u, Call p, u') \<in> edges fg"   and "proc_of fg v = p"   and "(v, M, P) \<in> S_cs fg 1"   and "q \<in># P"   and "(entry fg q, Ml, Me, h) \<in> RU_cs fg U"   and "(mon_n fg u \<union> mon fg p) \<inter> (Ml \<union> Me) = {}" shows "(u, mon fg p \<union> M, Ml \<union> Me, ah_update h (mon fg p, M) (Ml \<union> Me)) \<in> RU_cs fg U"
std_normal_distribution_odd_moments_abs: fixes k :: "nat" shows "LINT x|std_normal_distribution. \<bar>x\<bar> ^ (2 * k + 1) = sqrt (2 / pi) * 2 ^ k * fact k"
hfset_def: fixes a :: "hf" shows "hfset a = Abs_hf ` set_decode (Rep_hf a)"
norm_triangle_ineq2: 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 a :: "'a"   and b :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm a - norm b \<le> norm (minus a b)"
bv_to_int_msb0: fixes w1 :: "bit list" assumes "0 \<le> bv_to_int w1" shows "bv_msb w1 = \<zero>"
same_diag_trans: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat" assumes "local.same_diag A B"   and "local.same_diag B C" shows "local.same_diag A C"
simps_3: fixes t :: "tm"   and u :: "tm" shows "elts (Eats t u) = insert u (elts t)"
simps_1: fixes f :: "'c" shows "\<^bold>\<langle>f\<^bold>\<rangle>\<^sub>0\<^bold>\<down> = \<^bold>\<langle>f\<^bold>\<rangle>\<^sub>0"
\<Sigma>_protocol_def: shows "local.\<Sigma>_protocols_base.\<Sigma>_protocol \<equiv> local.\<Sigma>_protocols_base.completeness \<and> local.\<Sigma>_protocols_base.special_soundness \<and> local.\<Sigma>_protocols_base.HVZK"
supp_perm_eq: fixes x :: "'a"   and p :: "perm" assumes "supp x \<sharp>* p" shows "p \<bullet> x = x"
interior_of_mono: fixes S :: "'c set"   and T :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> T" shows "X interior_of S \<subseteq> X interior_of T"
initial_impl_def: fixes v0 :: "'q"   and I :: "'q \<Rightarrow> node_state option" shows "initial_impl v0 I \<equiv> GS_initial_impl I v0 (E `` {v0})"
ex_min_counterex: fixes I :: "'f set"   and N :: "'f set" assumes "\<not> I \<Turnstile> N" shows "\<exists>C\<in>N. \<not> I \<Turnstile> {C} \<and> (\<forall>D\<in>N. D < C \<longrightarrow> I \<Turnstile> {D})"
equivp_imp_transp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "equivp R" shows "transp R"
cont_fun: fixes x :: "'c" shows "cont (\<lambda>f. f x)"
cases: fixes a1 :: "char list exp list"   and a2 :: "heap \<times> sfields \<times> sheap"   and a3 :: "bool"   and a4 :: "char list exp list"   and a5 :: "heap \<times> sfields \<times> sheap"   and a6 :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>a1,a2,a3\<rangle> [\<rightarrow>] \<langle>a4,a5,a6\<rangle>" obtains e :: "char list exp" and s :: "heap \<times> sfields \<times> sheap" and b :: "bool" and e' :: "char list exp" and s' :: "heap \<times> sfields \<times> sheap" and b' :: "bool" and es :: "char list exp list" where "a1 = e # esb__" and "a2 = s__" and "a3 = bb__" and "a4 = e' # esb__" and "a5 = s'__" and "a6 = b'a__" and "P \<turnstile> \<langle>e,s__,bb__\<rangle> \<rightarrow> \<langle>e',s'__,b'a__\<rangle>"   | es :: "char list exp list" and s :: "heap \<times> sfields \<times> sheap" and b :: "bool" and es' :: "char list exp list" and s' :: "heap \<times> sfields \<times> sheap" and b' :: "bool" and v :: "val" where "a1 = Val v # es" and "a2 = s" and "a3 = b" and "a4 = Val v # es'" and "a5 = s'" and "a6 = b'" and "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>] \<langle>es',s',b'\<rangle>"
noStuck_startD': fixes t :: "('a, 'b) xstate"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'c, 'b) com option"   and c :: "('a, 'c, 'b) com"   and s :: "('a, 'b) xstate" assumes "t \<noteq> Stuck"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "s \<noteq> Stuck"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b" shows "(case LNil of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f1"
semilat_Def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semilat (A, r, f) = (order r A \<and> closed A f \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z))"
prime_factorization_eqI: fixes P :: "'a multiset"   and n :: "'a" assumes "\<And>p. p \<in># P \<Longrightarrow> prime p"   and "\<Prod>\<^sub># P = n" shows "prime_factorization n = P"
vimage_eq: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" shows "(a \<in> f -` B) = (f a \<in> B)"
denseinE: fixes T :: "'b set set"   and A :: "'b set"   and B :: "'b set" assumes "A densein\<^bsub>T\<^esub> B" obtains "B \<subseteq> closure\<^bsub>T\<^esub> A"
compl_set: fixes xs :: "'b list" shows "- set xs = List.coset xs"
that: fixes xl :: "'a" assumes "l = x # xs @ [xl]" shows "thesis"
coin_n: fixes dc :: "nat option \<times> bool list" shows "coin dc n = coin dc 0"
total_on_imp_Total_Restr: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "total_on A r" shows "Total (Restr r A)"
dsafe: shows "dsafe ($f fid1 (singleton (Var vid1)))"
fh: fixes t :: "real" assumes "dist x y \<le> t" shows "f t = h t"
perfect_choose_dist: fixes r :: "real"   and x :: "'c" assumes "0 < r" shows "\<exists>a. a \<noteq> x \<and> dist a x < r"
silent_move_empty_obs_slice: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')"   and "obs ms' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}" shows "obs ms \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}"
take_ltake: fixes i :: "nat"   and j :: "nat"   and xs :: "'d llist" assumes "i \<le> j" shows "take i (Co_Snapshot.ltake j xs) = Co_Snapshot.ltake i xs"
dual_dioid_one_zero: shows "class.dioid_one_zero (+) (\<odot>) (1::'a) (0::'a) (\<le>) (<)"
poly_gt_imp_poly_ge: fixes p :: "('c monom \<times> 'a) list"   and q :: "('c monom \<times> 'a) list" assumes "p >p q" shows "p \<ge>p q"
splitting_lemma_left: fixes C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b" assumes "exact_seq ([C, B, A], [g, f])"   and "f' \<in> hom B A"   and "\<And>x. x \<in> carrier A \<Longrightarrow> f' (f x) = x"   and "inj_on f (carrier A)"   and "g ` carrier B = carrier C" obtains H :: "'b set" and K :: "'b set" where "H \<lhd> B" and "K \<lhd> B" and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}" and "H <#>\<^bsub>B\<^esub> K = carrier B" and "f \<in> Group.iso A (subgroup_generated B H)" and "g \<in> Group.iso (subgroup_generated B K) C"
discrete_diagram_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" shows "discrete_diagram J C D \<equiv> category J \<and> diagram J C D \<and> discrete_diagram_axioms J"
lconcat_lappend_dist: fixes xs :: "'b llist llist"   and ys :: "'b llist llist" shows "lconcat\<cdot>(xs :++ ys) = lconcat\<cdot>xs :++ lconcat\<cdot>ys"
nhds_order: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.order_topology less_eq less open" shows "topological_space.nhds open x = inf (INF a\<in>ord.greaterThan less x. principal (ord.lessThan less a)) (INF a\<in>ord.lessThan less x. principal (ord.greaterThan less a))"
p_star_contr: fixes X :: "'a list set"   and Y :: "'a list set" shows "p_prod (p_star X) Y = (\<Union>n. p_prod (path_dioid.power X n) Y)"
ddv: shows "dv * cnj dv - nv * cnj nv = ddv * kk"
contfg: shows "continuous_on {a..b} (\<lambda>x. f (g x))"
simps_1: fixes ins :: "'b \<Rightarrow> 'a \<Rightarrow> 'a Heap"   and s :: "'a" shows "from_list_ga_aux ins [] s = return s"
prems_2: shows "ra setinterleaves ((trace_hide t A, trace_hide u A), C)"
assms_3: shows "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d"
sum_of_exp_transcendentalI: fixes xs :: "(complex \<times> complex) list" assumes "\<forall>(x, y)\<in>set xs. algebraic x \<and> algebraic y \<and> y \<noteq> 0"   and "\<exists>x\<in>fst ` set xs. x \<noteq> 0"   and "distinct (map snd xs)" shows "\<not> algebraic (\<Sum>(c, \<alpha>)\<leftarrow>xs. c * exp \<alpha>)"
wf_digraph_del_arc: fixes a :: "'b" shows "wf_digraph (del_arc a)"
left_closed: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). stk2.rel x y}" shows "a \<in> Sigma I \<FF>2"
distinct_2: fixes x21 :: "'b"   and x22 :: "'b"   and x23 :: "'c" shows "MkArr x21 x22 x23 \<noteq> Null"
simps_1: fixes f :: "nat \<Rightarrow> 'b" shows "annotate f SKIP = SKIP {f 0}"
preserves_iso: fixes f :: "'c" assumes "C.iso f" shows "D.iso (G f)"
comp_char: fixes g :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" shows "g \<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>] f = (if A2_B.seq g f then A2_B.MkArr (A2_B.Dom f) (A2_B.Cod g) (vertical_composite.map (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (A2_B.Map f) (A2_B.Map g)) else A2_B.null)"
simps_5: fixes a :: "Cooper.num"   and b :: "Cooper.num" shows "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
filter_top: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b" assumes "class.order_top less_eq less top" shows "ord.filter less_eq {top}"
guaranteeStripPair_split_conv: fixes c :: "'e \<Rightarrow> 'f set \<Rightarrow> 'd"   and f :: "'e"   and g :: "'f set" shows "(case guaranteeStripPair f g of (x, xa) \<Rightarrow> c x xa) = c f g"
ref_hd: fixes z :: "'a list"   and b :: "'a list"   and vs :: "'a list list"   and k :: "nat" assumes "z \<in> \<C>"   and "b \<in> \<C>"   and "z \<noteq> b"   and "vs \<in> lists (no_head_gen \<C> b)" shows "Ref \<C> (z \<cdot> b \<^sup>@ k) # vs = [z] \<cdot> [b] \<^sup>@ k \<cdot> Ref \<C> vs"
wf_PLUS: fixes xs :: "'b rexp list"   and \<Sigma> :: "'b set" shows "(atoms (PLUS xs) \<subseteq> \<Sigma>) = (\<forall>r\<in>set xs. atoms r \<subseteq> \<Sigma>)"
equiv_class_subset: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "equiv A r"   and "(a, b) \<in> r" shows "r `` {a} \<subseteq> r `` {b}"
coprime_cong_mult_poly: fixes a :: "'c poly"   and b :: "'c poly"   and m :: "'c poly"   and n :: "'c poly" assumes "[a = b] (mod m)"   and "[a = b] (mod n)"   and "coprime m n" shows "[a = b] (mod m * n)"
assms: shows "moebius_pt M z1 = moebius_pt M z2"
prim_impl_correct_2: fixes \<pi>i :: "(('a \<times> 'a) \<times> color) tree" assumes "G.invar (G.from_list l)"   and "prim_impl (G.from_list l) r = \<pi>i" shows "is_MST (G.\<alpha>w (G.from_list l)) (component_of (G.\<alpha>g (G.from_list l)) r) (P.\<alpha>_MST r \<pi>i)"
raw_match'_extend_map_preserve': fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs'' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'"   and "subs' \<subseteq>\<^sub>m subs''" shows "tsubstT T (convert_subs subs') = tsubstT T (convert_subs subs'')"
negGorbit_list_def: fixes smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and gs :: "'c list"   and T :: "'d \<Rightarrow> 'a"   and as :: "'d list" shows "aezfun_scalar_mult.negGorbit_list smult gs T as \<equiv> map (\<lambda>g. map (aezfun_scalar_mult.Gmult smult (- g) \<circ> T) as) gs"
psubstAbs_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" shows "A $[rho] \<equiv> asAbs pick A $[[pickE rho]]"
ref_arr_len_new: fixes s :: "Store"   and T :: "Arraytype"   and n :: "nat" shows "ref (arr_len (new s (new_array T n))) = new s (new_array T n)"
homeomorphic_maps_imp_homeomorphic_space: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "homeomorphic_maps X Y f g" shows "X homeomorphic_space Y"
wp1alternative_RS1: fixes p :: "('a, 'b) Combinators list" assumes "DenyAll \<in> set p" shows "wellformed_policy1_strong (removeShadowRules1 p)"
cong_l: fixes H :: "'a set" assumes "H \<in> normalfilters" shows "cong H = cong_l H"
indoc_appendE: fixes a :: "'item"   and u :: "'item list"   and v :: "'item list" assumes "indoc a (u @ v)" obtains "a \<in> set u" and "indoc a u"   | "a \<notin> set u" and "Ind {a} (set u)" and "indoc a v"
igFresh_termMOD: shows "igFresh termMOD = fresh"
distinct_pair_filter': fixes xs :: "('b \<times> 'c) list"   and a :: "'b"   and b :: "'c" assumes "distinct (map fst xs)"   and "(a, b) \<in> set xs" shows "map snd (filter (\<lambda>(x, y). x = a) xs) = [b]"
index_mat_swaprows_3: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_col (swaprows k l A) = dim_col A"
jordan_block_zero_pow: fixes n :: "nat"   and k :: "nat" shows "jordan_block n (0::'b) ^\<^sub>m k = mat n n (\<lambda>(i, j). if i \<le> j \<and> j - i = k then 1::'b else (0::'b))"
po_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x << y"   and "y << z" shows "x << z"
"5_1": shows "v = v\<^sub>1 @ [a] @ v\<^sub>2"
nat_set_cset_collapse_inj: shows "inj_on nat_set_cset_collapse {A. A \<noteq> {}\<^sub>c}"
assms_2: shows "\<not> (s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x))"
height_bound': fixes u :: "nat" assumes "u \<in> V" shows "relabel_effect f l u u \<le> 2 * card V - 1"
inner_dirsumD: fixes Ms :: "'a set list" assumes "add_independentS Ms" shows "(\<Oplus>M\<leftarrow>Ms. M) = (\<Sum>M\<leftarrow>Ms. M)"
lookup_pair_ko_ntm_def: fixes ko :: "'a nat_term_order" shows "lookup_pair_ko_ntm ko \<equiv> comparator.lookup_pair (nat_term_compare_inv ko)"
ActualE: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<A>\<phi> in v]" shows "[\<phi> in dw]"
rg_distrib: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "a \<cdot>\<^sub>r (b \<plusminus> c) = a \<cdot>\<^sub>r b \<plusminus> a \<cdot>\<^sub>r c"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "matched x1 x2 x3"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n ns n'' ns' n'. \<lbrakk>matched n ns n''; P n ns n''; n'' i-ns'\<rightarrow>\<^sub>d* n'\<rbrakk> \<Longrightarrow> P n (ns @ ns') n'"   and "\<And>n\<^sub>0 ns n\<^sub>1 p n\<^sub>2 ns' n\<^sub>3 n\<^sub>4 V a a'. \<lbrakk>matched n\<^sub>0 ns n\<^sub>1; P n\<^sub>0 ns n\<^sub>1; n\<^sub>1 -p\<rightarrow>\<^bsub>call\<^esub> n\<^sub>2; matched n\<^sub>2 ns' n\<^sub>3; P n\<^sub>2 ns' n\<^sub>3; n\<^sub>3 -p\<rightarrow>\<^bsub>ret\<^esub> n\<^sub>4 \<or> n\<^sub>3 -p:V\<rightarrow>\<^bsub>out\<^esub> n\<^sub>4; valid_edge a; a' \<in> get_return_edges a; sourcenode a = parent_node n\<^sub>1; targetnode a = parent_node n\<^sub>2; sourcenode a' = parent_node n\<^sub>3; targetnode a' = parent_node n\<^sub>4\<rbrakk> \<Longrightarrow> P n\<^sub>0 (ns @ n\<^sub>1 # ns' @ [n\<^sub>3]) n\<^sub>4"   and "\<And>n\<^sub>0 ns n\<^sub>1 p V n\<^sub>2 ns' n\<^sub>3 V' n\<^sub>4 a a'. \<lbrakk>matched n\<^sub>0 ns n\<^sub>1; P n\<^sub>0 ns n\<^sub>1; n\<^sub>1 -p:V\<rightarrow>\<^bsub>in\<^esub> n\<^sub>2; matched n\<^sub>2 ns' n\<^sub>3; P n\<^sub>2 ns' n\<^sub>3; n\<^sub>3 -p:V'\<rightarrow>\<^bsub>out\<^esub> n\<^sub>4; valid_edge a; a' \<in> get_return_edges a; sourcenode a = parent_node n\<^sub>1; targetnode a = parent_node n\<^sub>2; sourcenode a' = parent_node n\<^sub>3; targetnode a' = parent_node n\<^sub>4\<rbrakk> \<Longrightarrow> P n\<^sub>0 (ns @ n\<^sub>1 # ns' @ [n\<^sub>3]) n\<^sub>4" shows "P x1 x2 x3"
pratt: fixes p :: "nat" shows "prime p = (\<exists>c. Prime p \<in> set c \<and> valid_cert c)"
row_mat_of_rows_list: fixes b :: "nat"   and A :: "'b list list"   and nc :: "nat" assumes "b < length A"   and "\<forall>i<length A. length (A ! i) = nc" shows "row (mat_of_rows_list nc A) b = vec_of_list (A ! b)"
the_inv_into_f_f: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and x :: "'g" assumes "inj_on f A"   and "x \<in> A" shows "the_inv_into A f (f x) = x"
WTrtVal: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,E,h \<turnstile> Val v : T"
assms_1: shows "funpow_dist1 f x y < funpow_dist1 f x z"
is_top_sorted_append: fixes R :: "('b \<times> 'b) set"   and l1 :: "'b list"   and l2 :: "'b list" shows "is_top_sorted R (l1 @ l2) = (set l1 \<times> set l2 \<inter> (R\<^sup>*)\<inverse> = {} \<and> is_top_sorted R l1 \<and> is_top_sorted R l2)"
id: shows "(\<Sum>k = 0..<n. unit_vec n i $ k * v $ k) = unit_vec n i $ i * v $ i + (\<Sum>k\<in>{0..<n} - {i}. unit_vec n i $ k * v $ k)"
if_image_distrib: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and S :: "'b set" shows "(\<lambda>x. if P x then f x else g x) ` S = f ` (S \<inter> {x. P x}) \<union> g ` (S \<inter> {x. \<not> P x})"
sub_a_diff_d: shows "path_image p \<inter> closed_segment (a - complex_of_real d) (a - complex_of_real kde) \<subseteq> {a - complex_of_real d}"
int_obj: fixes G :: "prog"   and I :: "qtname" shows "G\<turnstile>Iface I\<preceq>Class Object"
nstut_before: fixes F :: "'a \<Rightarrow> 'b" shows "NSTUTINV $F"
distinct_81: fixes x4 :: "val"   and x6 :: "char list" shows "Val x4 \<noteq> Var x6"
d_fc_equals_dfc: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" assumes "deterministic P" shows "future_consistent P D R = d_future_consistent P D R"
indeps_5: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a" assumes "myss2 x y a z" shows "z \<bowtie> a"
cons3: fixes P :: "'a state \<Rightarrow> exp0 \<Rightarrow> 'a state \<times> (v, v) result \<Rightarrow> bool"   and s1 :: "'a state"   and e :: "exp0"   and s2 :: "'a state"   and v :: "v"   and es :: "exp0 list"   and s3 :: "'a state"   and err :: "v error_result" assumes "P s1 e (s2, Rval v)"   and "Big_Step_Unclocked_Single.evaluate_list P s2 es (s3, Rerr err)" shows "Big_Step_Unclocked_Single.evaluate_list P s1 (e # es) (s3, Rerr err)"
set_ConsD: fixes y :: "'c"   and x :: "'c"   and xs :: "'c list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
shift_compxEs2: fixes pc :: "nat"   and es :: "'a expr1 list"   and pc' :: "nat"   and d :: "nat" shows "shift pc (compxEs2 es pc' d) = compxEs2 es (pc' + pc) d"
left_inverse: fixes a :: "'ag" assumes "a \<in> U" shows "inverse a \<^bold>*\<^sub>o\<^sub>w a = \<^bold>1\<^sub>o\<^sub>w"
LIMSEQ_if_less: fixes i :: "nat"   and a :: "'c"   and b :: "'c" shows "(\<lambda>k. if i < k then a else b) \<longlonglongrightarrow> a"
rep_eq: fixes x :: "'a\<^sup>+"   and xa :: "'a\<^sup>+" shows "Rep_pos (x + xa) = Rep_pos x + Rep_pos xa"
intPF_def: shows "intPF \<equiv> InfModel.intPF parOf intTI intPI"
fold_invariant: fixes xs :: "'c list"   and Q :: "'c \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and s :: "'d"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x"   and "P s"   and "\<And>x s. \<lbrakk>Q x; P s\<rbrakk> \<Longrightarrow> P (f x s)" shows "P (fold f xs s)"
N_comp: fixes x :: "'a"   and y :: "'a" shows "N x * y = - (- x * top) \<sqinter> y"
one_sheaf_spec_def: fixes U :: "nat set set" shows "cring0.one_sheaf_spec U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.one_rel ({0}\<setminus>\<pp>) {0} ring0.subtraction ring0.subtraction 0 0"
ImplE: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and b :: "('a, 'b) form" assumes "G \<turnstile> Impl a b"   and "G \<turnstile> a" shows "G \<turnstile> b"
simps_131: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x101 :: "16 word"   and x102 :: "16 word" shows "(case L4Src x101 x102 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f10 x101 x102"
swap_restrict: fixes A :: "'d set"   and B :: "'e set"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "finite A"   and "finite B" shows "(\<Sum>x\<in>A. sum_0 (g x) {y \<in> B. R x y}) = (\<Sum>y\<in>B. \<Sum>x | x \<in> A \<and> R x y. g x y)"
subdegree_def: fixes f :: "'a fps" shows "subdegree f = (if f = 0 then 0 else LEAST n. fps_nth f n \<noteq> (0::'a))"
finite_cone_def: fixes Vs :: "'a vec set" shows "finite_cone Vs = {b. \<exists>c. nonneg_lincomb c (if finite Vs then Vs else {}) b}"