hcju/isabelleps huggingface dataset:
min_diff_distrib_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "minus (ord.min less_eq x y) z = ord.min less_eq (minus x z) (minus y z)"
span_Basis: shows "module_on.span (*\<^sub>R) basis = sub.tangent_space p"
exec_meth_xcpt: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" shows "exec_meth ci P ins xt t h (stk, loc, pc, \<lfloor>xcp\<rfloor>) ta h (stk', loc', pc', xcp') = (\<exists>d. match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor> \<and> ta = \<lbrace>\<rbrace> \<and> stk' = Addr xcp # drop (length stk - d) stk \<and> loc' = loc \<and> xcp' = None \<and> d \<le> length stk)"
ExcessTable'_def: fixes g :: "graph" shows "ExcessTable' g \<equiv> ExcessTable_cont (ExcessAt g)"
nocoder: fixes x :: "\<kappa>" shows "[[\<lparr>O!,x\<rparr> \<^bold>\<rightarrow> \<^bold>\<not>(\<^bold>\<exists>F. \<lbrace>x,F\<rbrace>)]]"
simps_1: fixes v :: "bool"   and res :: "bool"   and S :: "'av Abs_State.st option" shows "bfilter (Bc v) res S = (if v = res then S else None)"
reads_def: fixes S :: "('l \<Rightarrow> 'l \<Rightarrow> bool) set"   and getter :: "('l, 'm, 'n) prog"   and h :: "'l"   and h' :: "'l" shows "reads S getter h h' = ((\<forall>P\<in>S. reflp P \<and> transp P) \<and> ((\<forall>P\<in>S. P h h') \<longrightarrow> preserved getter h h'))"
r_split1_not_none: fixes r :: "'b wordinterval" assumes "\<not> wordinterval_empty r" shows "fst (wordinterval_CIDR_split1 r) \<noteq> None"
spectral_radius_ev_def: fixes A :: "((complex, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "HMA_Connect.spectral_radius A = Max (cmod ` Collect (eigen_value A))"
Cable_def: fixes n :: "nat"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list" shows "Cable n nSend recv \<equiv> CorrectSheaf n \<and> (inf_disj n nSend \<longrightarrow> Broadcast n nSend recv)"
nub_body''_nub_body'''_eq: shows "nub_body'' = nub_body''' oo unwrap oo wrap"
lowner_le_traceD: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and \<rho> :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "\<rho> \<in> carrier_mat n n"   and "A \<le>\<^sub>L B"   and "partial_density_operator \<rho>" shows "trace (A * \<rho>) \<le> trace (B * \<rho>)"
nth_default_eq_iff: fixes dflt :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(nth_default dflt xs = nth_default dflt ys) = (strip_while ((=) dflt) xs = strip_while ((=) dflt) ys)"
inf_inf_implies: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<sqinter> (x \<sqinter> y \<leadsto> x) = z"
filter_False: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> P x" shows "filter P xs = []"
fully_faithful_functor_L: shows "fully_faithful_functor Left_a.comp Left_a.comp Left_a.L"
kD_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and nip :: "nat"   and v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "kD (rt(nip \<mapsto> v)) = insert nip (kD rt)"
keys_to_list_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys_to_list p = pps_to_list (keys p)"
foldr_Cons: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
if_distribR: fixes b :: "bool"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(if b then f else g) x = (if b then f x else g x)"
coprime_lead_coeff_factor_1: fixes m :: "int"   and p :: "int"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod_2 m"   and "prime p"   and "comm_monoid_mult_class.coprime (lead_coeff (f * g)) p" shows "comm_monoid_mult_class.coprime (lead_coeff f) p"
alg_dvd_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x alg_dvd y"   and "y alg_dvd z" shows "x alg_dvd z"
1: shows "bprv (PP \<langle>imp (\<phi>L \<phi>) (imp (PP \<langle>\<phi>L \<phi>\<rangle>) \<phi>)\<rangle>)"
symvars_of_adapt_bexp: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "'c \<Rightarrow> nat" shows "Bexp.vars (adapt_bexp e s) \<subseteq> symvars s"
path_image_join_subset: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" shows "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2"
iso_runit': fixes f :: "'c" assumes "C.ide f" shows "C.iso \<r>\<^sub>C\<^sup>-\<^sup>1[f]"
RedTryCatch: fixes s :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and b :: "bool" assumes "hp s a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>try Throw a catch(C V) e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>{V:Class C; V:=addr a;; e\<^sub>2},s,b\<rangle>"
cbigD: fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and combine :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and collect_id :: "'c"   and \<sigma>' :: "'b"   and coll' :: "'c"   and collect :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c"   and P :: "'a"   and \<sigma> :: "'b" assumes "CollectionSemantics small endset combine collect_id"   and "(\<sigma>', coll') \<in> CollectionSemantics.cbig small endset collect combine collect_id P \<sigma>" shows "\<exists>n. (\<sigma>', coll') \<in> CollectionSemantics.csmall_nstep small collect combine collect_id P \<sigma> n \<and> \<sigma>' \<in> endset"
sip_in_kD: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:7} \<union> {PAodv-:5} \<union> {PRrep-:0..PRrep-:4} \<union> {PRreq-:0..PRreq-:3} \<longrightarrow> sip \<xi> \<in> kD (rt \<xi>))"
set1_F_map: fixes l1 :: "'n \<Rightarrow> 'x"   and l2 :: "'o \<Rightarrow> 'y"   and l3 :: "'p \<Rightarrow> 'z" shows "set1_F \<circ> mapl_F l1 l2 l3 = (`) l1 \<circ> set1_F"
split_list_last_prop_iff: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>set xs. P x) = (\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z\<in>set zs. \<not> P z))"
enat_cosplit_asm: fixes P :: "'a \<Rightarrow> bool"   and c :: "'a"   and d :: "enat \<Rightarrow> 'a"   and n :: "enat" shows "P (case n of 0 \<Rightarrow> c | eSuc x \<Rightarrow> d x) = (\<not> (n = 0 \<and> \<not> P c \<or> (\<exists>m. n = eSuc m \<and> \<not> P (d m))))"
sorted_wrt01: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "length xs \<le> 1" shows "sorted_wrt P xs"
set_times_mono: fixes C :: "'b set"   and D :: "'b set"   and a :: "'b" assumes "C \<subseteq> D" shows "a *o C \<subseteq> a *o D"
apath_Nil_iff: fixes u :: "'a"   and v :: "'a" shows "apath u [] v = (u = v \<and> u \<in> verts G)"
linepath_le_1: fixes a :: "'c"   and b :: "'c"   and u :: "'c" assumes "a \<le> (1::'c)"   and "b \<le> (1::'c)"   and "(0::'c) \<le> u"   and "u \<le> (1::'c)" shows "((1::'c) - u) * a + u * b \<le> (1::'c)"
distinct_9: fixes x21 :: "char list"   and x22 :: "expr"   and x31 :: "com"   and x32 :: "com" shows "x21:=x22 \<noteq> x31;; x32"
poly_mult_degree_eq: fixes K :: "'a list set"   and p1 :: "'a list list"   and p2 :: "'a list list" assumes "subring K (poly_ring R)"   and "polynomial\<^bsub>poly_ring R\<^esub> K p1"   and "polynomial\<^bsub>poly_ring R\<^esub> K p2" shows "degree (UP.poly_mult p1 p2) = (if p1 = [] \<or> p2 = [] then 0 else degree p1 + degree p2)"
get_tag_name_impl: shows "get_tag_name = a_get_tag_name"
append_eqvt: fixes pi :: "('b \<times> 'b) list"   and l1 :: "'a list"   and l2 :: "'a list" shows "pi \<bullet> (l1 @ l2) = pi \<bullet> l1 @ pi \<bullet> l2"
domain_positive: fixes x :: "brick"   and y :: "brick list" assumes "0 < domain_block [x] \<or> 0 < domain_block y" shows "0 < domain_block (x # y)"
b: shows "\<pi>\<^sup>* y \<in> carrier \<Y>"
closed_affine_hull: fixes S :: "'b set" shows "closed (affine hull S)"
fix_clock_2: fixes s1 :: "'c state"   and s2 :: "'a state"   and x :: "'b"   and s :: "'a state" assumes "fix_clock s1 (s2, x) = (s, x)" shows "clock s \<le> clock s2"
Lim_bounded: fixes f :: "nat \<Rightarrow> 'c"   and l :: "'c"   and M :: "nat"   and C :: "'c" assumes "f \<longlonglongrightarrow> l"   and "\<forall>n\<ge>M. f n \<le> C" shows "l \<le> C"
ltl_const_equiv_implies_prop_equiv_2: fixes \<phi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>C false\<^sub>n = \<phi> \<sim>\<^sub>P false\<^sub>n"
lookup_pair_map_pair: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and f :: "'a \<times> 'd \<Rightarrow> 'a \<times> 'e"   and k :: "'a"   and v :: "'d" assumes "oalist_inv_raw x xs"   and "snd (f (k, 0::'d)) = (0::'e)"   and "\<And>a b. key_compare (rep_key_order x) (fst (f a)) (fst (f b)) = key_compare (rep_key_order x) (fst a) (fst b)" shows "lookup_pair x (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_pair x xs k))"
DSourcesS15: shows "DSources level2 sS15 = {sS14}"
t_mtf_def: fixes n :: "nat" shows "t_mtf n = int (index (s_mtf n) (rs ! n) + 1)"
diff_cancel: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "k + m - (k + n) = m - n"
A_Rep: fixes b :: "'al"   and b' :: "'ar" shows "A b b' = (b = Rep_a b')"
distinct_1: fixes x1 :: "real mpoly"   and x2 :: "real mpoly" shows "Less x1 \<noteq> Eq x2"
i_join_i_shrink_iT_Plus_iT_Div_mod_subst: fixes k :: "nat"   and I :: "nat set"   and A :: "nat set"   and B :: "nat set"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k"   and "I \<noteq> {}"   and "\<forall>x\<in>I. x mod k = 0"   and "A = I \<oplus> k - 1"   and "B = I \<oslash> k" shows "f \<longmapsto> k \<Join>\<^sub> A = f \<div> k \<Join>\<^sub> B"
less_imp_erf_real_less: fixes a :: "real"   and b :: "real" assumes "a < b" shows "erf a < erf b"
ereal_less_1: fixes r :: "real" shows "(ereal r < 0) = (r < 0)"
no_trailing_append_Cons: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "no_trailing P (xs @ y # ys) = no_trailing P (y # ys)"
orderI: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a = a \<union># b" shows "b \<subseteq># a"
exec': shows "exec_meth_d (compP2 P) (compE2 e @ compEs2 es) (compxE2 e 0 0 @ shift (length (compE2 e)) (stack_xlift (length [v]) (compxEs2 es 0 0))) t h (stk @ [v], loc, length (compE2 e) + pc, xcp) ta h' (stk', loc', pc', xcp')"
section_def: fixes f :: "'d" shows "D.section f \<equiv> \<exists>g. D.ide (g \<cdot>\<^sub>D f)"
reduced_calc_is_calc: shows "calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
SHOmsgVectors_def: fixes A :: "('a, 'c, 'b, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'a"   and cfg :: "'a \<Rightarrow> 'c"   and HOp :: "'a set"   and SHOp :: "'a set" shows "SHOmsgVectors A r p cfg HOp SHOp \<equiv> {\<mu>. (\<forall>q. (q \<in> HOp) = (\<mu> q \<noteq> None)) \<and> (\<forall>q. q \<in> SHOp \<inter> HOp \<longrightarrow> \<mu> q = Some (sendMsg A r q p (cfg q)))}"
wt_ge_\<epsilon>: fixes s :: "('s, 'v) tm" shows "\<epsilon> \<le> wt s"
bij_betw_apply: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and a :: "'a" assumes "bij_betw f A B"   and "a \<in> A" shows "f a \<in> B"
rel_optionT'_phantom: fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_optionT' A = rel_optionT' \<top>"
simps_4: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "summands (a{b}.P) = {a{b}.P}"
Aw'_def: shows "Aw' = ten2mat Aw"
extremally_disconnected_def: fixes X :: "'b set" shows "extremally_disconnected X = (K \<circ> I \<circ> K = I \<circ> K)"
square_final_mat: shows "square_mat (get_matrix (reduce_system p (qs, M_mat signs subsets, subsets, signs)))"
prod_relI: fixes a :: "'c"   and a' :: "'d"   and R1 :: "('c \<times> 'd) set"   and b :: "'e"   and b' :: "'f"   and R2 :: "('e \<times> 'f) set" assumes "(a, a') \<in> R1"   and "(b, b') \<in> R2" shows "((a, b), a', b') \<in> R1 \<times>\<^sub>r R2"
init_is_traversed: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "s \<in> traversed_states s as"
eexp_special_values_iff_1: fixes x :: "ereal" shows "(eexp x = 0) = (x = - \<infinity>)"
hcomp_ide_dom: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd" assumes "natural_transformation A B F G \<tau>" shows "\<tau> \<circ> identity_functor.map A = \<tau>"
simps_2: fixes ExcessAtPG :: "nat \<Rightarrow> nat"   and v :: "nat"   and vs :: "nat list" shows "ExcessTable_cont ExcessAtPG (v # vs) = (let vi = ExcessAtPG v in if 0 < vi then (v, vi) # ExcessTable_cont ExcessAtPG vs else ExcessTable_cont ExcessAtPG vs)"
assms: shows "A \<subseteq> U\<^sub>V\<^sub>S"
\<C>_is_code: fixes xs :: "'a list list"   and ys :: "'a list list" assumes "xs \<in> lists \<C>"   and "ys \<in> lists \<C>"   and "concat xs = concat ys" shows "xs = ys"
E_def: shows "E = mat_of_rows (dim_col A) (map (Matrix.row C) [2..<dim_row A])"
return_needs_call: fixes a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<exists>!a'. valid_edge a' \<and> (\<exists>Q r fs. kind a' = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs) \<and> a \<in> get_return_edges a'"
phi_not_top: fixes pc :: "nat" assumes "wtl ins c 0 s\<^sub>0 \<noteq> \<top>"   and "pc < length ins" shows "\<tau>s ! pc \<noteq> \<top>"
Scons_inject2: fixes M :: "('c, 'd) Old_Datatype.node set"   and N :: "('c, 'd) Old_Datatype.node set"   and M' :: "('c, 'd) Old_Datatype.node set"   and N' :: "('c, 'd) Old_Datatype.node set" assumes "Scons M N = Scons M' N'" shows "N = N'"
abs_rep_a: fixes y :: "'ar" shows "y = Abs_a (Rep_a y)"
sublist_append_leftI: fixes xs :: "'a list"   and ps :: "'a list" shows "sublist xs (ps @ xs)"
simps: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and r :: "Vars.var list"   and c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" shows "conf_transform (ae, ce, a, as, r) c = add_dummies_conf r (a_transform (ae, a, as) (restr_conf (- set r) c))"
least_lselect_llength: fixes w :: "'b llist"   and s :: "nat set" assumes "\<not> lnull (lselect s w)" shows "enat (least s) < llength w"
l1_inv4_trans: shows "{l1_inv4} TS.trans l1 {> l1_inv4}"
deriv_bodies: shows "\<forall>p\<in>set Procs. \<forall>\<sigma> Z. \<Gamma>,\<Theta> \<union> (\<Union>q\<in>set Procs. \<Union>Z. {(P q Z \<inter> {s. ((s, q), \<sigma>, p) \<in> r}, q, Q q Z, A q Z)})\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"
incoming_alt: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat" shows "Graph.incoming c u = (\<lambda>v. (v, u)) ` (Graph.E c)\<inverse> `` {u}"
no_loops: fixes e :: "'a \<times> 'a" assumes "e \<in> arcs (with_proj G)" shows "tail (with_proj G) e \<noteq> head (with_proj G) e"
integral_divide_zero: fixes M :: "'e measure"   and f :: "'e \<Rightarrow> 'd"   and c :: "'d" shows "LINT x|M. f x / c = integral\<^sup>L M f / c"
norm_empty_bv_to_nat_zero: fixes w :: "bit list" assumes "norm_unsigned w = []" shows "bv_to_nat w = 0"
fbd_ifbd_galois: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and f :: "'c \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "(bd\<^sup>-\<^sub>\<F> \<phi> = f) = (bd\<^sub>\<F> f = \<phi>)"
is_inf_uniq: fixes x :: "'b"   and y :: "'b"   and inf :: "'b"   and inf' :: "'b" assumes "is_inf x y inf"   and "is_inf x y inf'" shows "inf = inf'"
diff_funD: fixes x :: "'a" assumes "x \<in> src.carrier" shows "\<exists>c1\<in>src.atlas. x \<in> domain c1 \<and> k-smooth_on (codomain c1) (f \<circ> inv_chart c1)"
\<tau>Red1_conv: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and ex :: "'addr expr1 \<times> 'addr val list"   and exs :: "('addr expr1 \<times> 'addr val list) list"   and ex' :: "'addr expr1 \<times> 'addr val list"   and exs' :: "('addr expr1 \<times> 'addr val list) list" shows "\<tau>Red1g uf P t h (ex, exs) (ex', exs') = (uf,P,t \<turnstile>1 \<langle>ex/exs,h\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>ex'/exs',h\<rangle> \<and> \<tau>Move1 P h (ex, exs))"
ty_ex_pres: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and fs :: "local_state \<times> global_state"   and a :: "action"   and fs' :: "local_state \<times> global_state" assumes "ty_fs \<Gamma> fs"   and "la_ex fs a = Some fs'" shows "ty_fs \<Gamma> fs'"
productive_imp_Interp: fixes C :: "'a literal multiset" assumes "production C \<noteq> {}" shows "Interp C \<Turnstile> C"
general_product_eq: fixes F :: "i \<Rightarrow> bool" shows "(\<pi>x. F x) = (\<sigma>x. \<forall>y. F y \<longrightarrow> P x y)"
diagonal_to_Smith_row_i_dvd_jj: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and a :: "'b"   and i :: "nat"   and A :: "(('a, 'c) vec, 'b) vec"   and b :: "'c" assumes "is_bezout_ext bezout"   and "to_nat a \<in> {i..<min (nrows A) (ncols A)}"   and "to_nat a = to_nat b" shows "Diagonal_to_Smith_row_i A i bezout $ from_nat i $ from_nat i dvd Diagonal_to_Smith_row_i A i bezout $ a $ b"
subset_Idl_subset: fixes I :: "'a set"   and H :: "'a set" assumes "I \<subseteq> carrier R"   and "H \<subseteq> I" shows "Idl H \<subseteq> Idl I"
one_circ_L: shows "(1::'a)\<^sup>\<circ> * L = L"
assms: shows "(all_shared xs \<union> all_unshared xs \<union> all_acquired xs) \<inter> (all_shared ys \<union> all_unshared ys \<union> all_acquired ys) = {}"
sup_aci_1: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
standard_uniqueness_automorphs: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and C :: "'a set" assumes "ChamberComplexAutomorphism X f"   and "ChamberComplexAutomorphism X g"   and "chamber C"   and "fun_eq_on f g C" shows "f = g"
list_all2_map1: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and as :: "'f list"   and bs :: "'e list" shows "list_all2 P (map f as) bs = list_all2 (\<lambda>x. P (f x)) as bs"
intro: fixes M :: "'c \<Rightarrow> 'd measure" assumes "\<And>i. prob_space (M i)" shows "product_prob_space_axioms M"
does_not_communicate_with_gateway_def: fixes u :: "'dom_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "does_not_communicate_with_gateway u execs \<equiv> \<forall>a. a \<in> actions_in_execution (execs u) \<longrightarrow> (\<forall>v. intermediary v u \<longrightarrow> v \<notin> involved (Some a))"
\<G>\<F>_subset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "\<G>\<F> \<psi> w \<subseteq> \<G>\<F> \<phi> w"
eq_OO: fixes R :: "'p \<Rightarrow> 'q \<Rightarrow> bool" shows "(=) OO R = R"
mult_le_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a b) zero = (less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b)"
sup: fixes f :: "real \<Rightarrow> 'f"   and F1 :: "real filter"   and g :: "real \<Rightarrow> 'f"   and F2 :: "real filter" assumes "f \<in> L F1 g"   and "f \<in> L F2 g" shows "f \<in> L (sup F1 F2) g"
dpo_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "dpo le \<equiv> (\<forall>x. le x x) \<and> (\<forall>x y. le x y \<longrightarrow> le y x \<longrightarrow> x = y) \<and> (\<forall>x y z. le x y \<longrightarrow> le y z \<longrightarrow> le x z)"
le_Suc: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less a b" shows "less_eq (plus a one) b"
rSum1: assumes "\<Psi> \<rhd> P \<longmapsto> Rs"   and "guarded P" shows "Prop"
FL_valid_eqvt: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and effect_apply :: "'e \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'d \<times> 'e fs_set \<times> 'e \<Rightarrow> 'e fs_set"   and P :: "'b"   and x :: "('f, 'c, 'd, 'e) formula"   and p :: "perm" assumes "effect_nominal_ts satisfies transition effect_apply L"   and "effect_nominal_ts.FL_valid satisfies transition effect_apply P x" shows "effect_nominal_ts.FL_valid satisfies transition effect_apply (p \<bullet> P) (p \<bullet> x)"
confTs_hext: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and loc :: "val list"   and LT :: "ty err list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"   and "h \<unlhd> h'" shows "P,h' \<turnstile> loc [:\<le>\<^sub>\<top>] LT"
generate_in_carrier: fixes H :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set set"   and h :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "H \<subseteq> carrier (homology_group p X)"   and "h \<in> generate (homology_group p X) H" shows "h \<in> carrier (homology_group p X)"
in_inv_image: fixes x :: "'e"   and y :: "'e"   and r :: "('f \<times> 'f) set"   and f :: "'e \<Rightarrow> 'f" shows "((x, y) \<in> inv_image r f) = ((f x, f y) \<in> r)"
bind_simp: fixes x :: "'a + 'c"   and y :: "'c \<Rightarrow> 'a + 'b"   and r :: "'b" assumes "x \<bind> y = Inr r" shows "y (projr x) = Inr r"
ordIso_finite_Field: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" assumes "r =o s" shows "finite (Field r) = finite (Field s)"