hcju/isabelleps huggingface dataset:
args_to_set_alt: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "args_to_set (gs, bs, ps) = fst ` set gs \<union> fst ` set bs \<union> fst ` fst ` set ps \<union> fst ` snd ` set ps"
deg_le_sect_deg_le_sect_pp: fixes xs :: "'x list"   and d :: "nat" shows "map PP (pp_pm.punit.pps_to_list (deg_le_sect (set xs) d)) = deg_le_sect_pp xs d"
type_I_chain_horiz_bound_valid: shows "\<forall>(k, \<gamma>)\<in>two_chain_horizontal_boundary two_chain. valid_path \<gamma>"
expand_conj_in_set: fixes w :: "nat"   and Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list" shows "{x. x \<le> Suc w \<and> Rr (xs @ [x])} = (if Rr (xs @ [Suc w]) then insert (Suc w) {x. x \<le> w \<and> Rr (xs @ [x])} else {x. x \<le> w \<and> Rr (xs @ [x])})"
Sup_pair_ow_def: fixes U\<^sub>1 :: "'a set"   and Sup\<^sub>1 :: "'a set \<Rightarrow> 'a"   and U\<^sub>2 :: "'b set"   and Sup\<^sub>2 :: "'b set \<Rightarrow> 'b" shows "Sup_pair_ow U\<^sub>1 Sup\<^sub>1 U\<^sub>2 Sup\<^sub>2 \<equiv> Sup_ow U\<^sub>1 Sup\<^sub>1 \<and> Sup_ow U\<^sub>2 Sup\<^sub>2"
intro: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and W :: "'d set" assumes "vectorspace K V"   and "LinearCombinations.submodule K W V" shows "subspace K W V"
Var: fixes l :: "char list \<Rightarrow> val option"   and V :: "char list"   and v :: "val"   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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "l V = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>Var V,(h, l)\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"
Nil\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" shows "P \<turnstile>\<^sub>1 \<langle>[],s\<rangle> [\<Rightarrow>] \<langle>[],s\<rangle>"
sin_squared_eq: fixes x :: "'b" shows "(sin x)\<^sup>2 = (1::'b) - (cos x)\<^sup>2"
assms_5: fixes s :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set" assumes "s \<in> set ps" shows "phull.subspace s"
cond_def: fixes a :: "nat \<Rightarrow> 'b vec"   and m :: "nat"   and n :: "nat"   and b :: "'b vec"   and I :: "nat set"   and I' :: "nat set"   and l :: "'b vec \<Rightarrow> 'b"   and c :: "'b vec"   and h :: "nat"   and k :: "nat" assumes "a ` {0..<m} \<subseteq> carrier_vec n"   and "inj_on a {0..<m}"   and "b \<in> carrier_vec n"   and "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (a ` {0..<m}) = carrier_vec n" shows "gram_schmidt.cond n a b m I I' l c h k \<equiv> b = module.lincomb (module_vec TYPE('b) n) l (a ` I) \<and> h \<in> I \<and> l (a h) < (0::'b) \<and> (\<forall>h'. h' \<in> I \<longrightarrow> h' < h \<longrightarrow> (0::'b) \<le> l (a h')) \<and> c \<in> carrier_vec n \<and> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (a ` (I - {h})) = {x \<in> carrier_vec n. c \<bullet> x = (0::'b)} \<and> c \<bullet> b < (0::'b) \<and> k < m \<and> c \<bullet> a k < (0::'b) \<and> (\<forall>k'<k. (0::'b) \<le> c \<bullet> a k') \<and> I' = insert k (I - {h})"
Derivs_def: fixes xs :: "'a list"   and A :: "'a list set" shows "Derivs xs A = {ys. xs @ ys \<in> A}"
iT_Plus_neg_iAll_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I \<oplus>- k. P t) = (\<box> t I \<down>\<ge> k. P (t - k))"
Nat_induct: fixes P :: "nat discr Box \<Rightarrow> bool"   and x :: "nat discr Box" assumes "P \<bottom>"   and "P 0"   and "\<And>n. \<lbrakk>n \<noteq> \<bottom>; P n\<rbrakk> \<Longrightarrow> P (n + 1)" shows "P x"
Abs_inverse: fixes y :: "'c llist" assumes "y \<in> {xs. \<not> lfinite xs}" shows "llist_of_stream (stream_of_llist y) = y"
is_processT6_S2: fixes X :: "'a event set"   and P :: "'a process" assumes "tick \<notin> X"   and "[tick] \<in> T P" shows "([], X) \<in> F P"
lfp_lowerbound: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq (f A) A" shows "less_eq (complete_lattice.lfp Inf less_eq f) A"
of_rat_Real: fixes x :: "rat" shows "real_of_rat x = Real.Real (\<lambda>n. x)"
span_not_univ_subset_hyperplane: fixes S :: "'a set" assumes "span S \<noteq> UNIV" shows "\<exists>a. a \<noteq> (0::'a) \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
wls_swapAbs_commute: fixes zs :: "'varSort"   and zs' :: "'varSort"   and x :: "'var"   and y :: "'var"   and x' :: "'var"   and y' :: "'var"   and us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "zs \<noteq> zs' \<or> {x, y} \<inter> {x', y'} = {}"   and "wlsAbs (us, s) A" shows "A $[x \<and> y]_zs $[x' \<and> y']_zs' = A $[x' \<and> y']_zs' $[x \<and> y]_zs"
I_def: shows "I = (\<lambda>N K e. indicat_real (- B N K e))"
lex_prodp_wfP: fixes r1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "wfP r1"   and "wfP r2" shows "wfP (lex_prodp r1 r2)"
Times_eq_image_sum: fixes S :: "'d set"   and T :: "'e set" shows "S \<times> T = {u + v |u v. u \<in> (\<lambda>x. (x, 0::'e)) ` S \<and> v \<in> Pair (0::'d) ` T}"
prems_1: shows "inv_max_hi \<langle>l, (a, m), r\<rangle>"
is_idirprod_imp_compl_fam: fixes A :: "'a set"   and S :: "'c \<Rightarrow> 'a set"   and I :: "'c set" assumes "is_idirprod A S I" shows "compl_fam S I"
cupcake_single_complete0: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and res :: "'b state \<times> (v, v) result" assumes "Big_Step_Unclocked_Single.evaluate env s e res"   and "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "cupcake_evaluate_single env e (snd res)"
assms: fixes r :: "'a" assumes "r dvd p"   and "r dvd q" shows "r dvd (1::'a)"
greater: shows "r'\<^sub>2 < r'\<^sub>1"
Cons_eq_append_conv: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
LIM_offset_zero_iff: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and L :: "'c" assumes "NO_MATCH (0::'a) a" shows "f \<midarrow>a\<rightarrow> L = (\<lambda>h. f (a + h)) \<midarrow>0::'b\<rightarrow> L"
ctr: fixes r :: "real"   and x :: "complex" shows "r *\<^sub>R x = Complex (r * Re x) (r * Im x)"
primitive_rootP_def: fixes x :: "'b list"   and r :: "'b list" shows "x \<in>\<^sub>p r * = (x \<noteq> \<epsilon> \<and> x \<in> r* \<and> primitive r)"
int_of_uint32_shift: fixes n :: "uint32"   and k :: "nat" shows "int_of_uint32 (n >> k) = int_of_uint32 n div 2 ^ k"
update_convs_2: fixes cy' :: "int \<Rightarrow> int"   and cx :: "int"   and cy :: "int"   and more :: "'a" shows "cy_update cy' \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = \<lparr>cx = cx, cy = cy' cy, \<dots> = more\<rparr>"
isomorphic_def: fixes a :: "'c \<times> 'c"   and a' :: "'c \<times> 'c" shows "C.VV.isomorphic a a' = (\<exists>f. C.VV.in_hom f a a' \<and> C.VV.iso f)"
ldropn_fixp_case_conv: fixes f :: "'c llist \<Rightarrow> 'd \<Rightarrow> 'c llist" shows "(\<lambda>xs. case xs of LNil \<Rightarrow> \<lambda>_. LNil | LCons x xs \<Rightarrow> \<lambda>n. if n = (0::'d) then LCons x xs else f xs (n - (1::'d))) = (\<lambda>xs n. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> if n = (0::'d) then LCons x xs else f xs (n - (1::'d)))"
lset_induct: fixes x :: "'c"   and xs :: "'c llist"   and P :: "'c llist \<Rightarrow> bool" assumes "x \<in> lset xs"   and "\<And>xs. P (LCons x xs)"   and "\<And>x' xs. \<lbrakk>x \<in> lset xs; x \<noteq> x'; P xs\<rbrakk> \<Longrightarrow> P (LCons x' xs)" shows "P xs"
balanced: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "int (card ps) = 2" shows "int (\<B> index ps) = 1"
map2_map: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'c"   and as :: "'d list" shows "map2 f (map g [0..<length as]) as = map (\<lambda>i. f (g i) (as ! i)) [0..<length as]"
check_matrix_complexity_def: fixes A :: "real mat"   and cp :: "real poly"   and d :: "nat" shows "check_matrix_complexity A cp d = (count_roots_above cp 1 = 0 \<and> (poly cp 1 = 0 \<longrightarrow> (let ord = order 1 cp in d + 1 < ord \<longrightarrow> kernel_dim ((A - 1\<^sub>m (dim_row A)) ^\<^sub>m (d + 1)) = ord)))"
C_1: shows "limsup (\<lambda>x. ereal (u x)) < C"
single_valuedp_def: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "single_valuedp r = (\<forall>x y. r x y \<longrightarrow> (\<forall>z. r x z \<longrightarrow> y = z))"
less_eq_multiset_total: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "\<not> M \<le> N" shows "N \<le> M"
predicate2I: fixes P :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Q :: "'j \<Rightarrow> 'k \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
GAUSS_def: fixes p :: "nat"   and a :: "int" shows "GAUSS p a \<equiv> (prime p \<and> 2 < p) \<and> [a \<noteq> 0] (mod int p) \<and> 0 < a"
onode_sos_net_states: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and s :: "'b net_state"   and a :: "'c node_action"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and s' :: "'b net_state"   and S :: "(((nat \<Rightarrow> 'a) \<times> 'b) \<times> 'c seq_action \<times> (nat \<Rightarrow> 'a) \<times> 'b) set" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> onode_sos S" shows "\<exists>i \<zeta> R \<zeta>' R'. s = NodeS i \<zeta> R \<and> s' = NodeS i \<zeta>' R'"
simps_1: fixes A :: "char list"   and X :: "ntrm set" shows "NEGc (PR A) X = AXIOMSc (PR A) \<union> BINDINGc (PR A) X"
tensor\<^sub>S_def: fixes f :: "'c local.term set"   and g :: "'c local.term set" shows "f \<otimes>\<^sub>S g \<equiv> \<F>C.D (\<F>C.tensor f g)"
NoSuchFieldC_def: shows "NoSuchFieldC \<equiv> (NoSuchFieldError, Object, [], [])"
elims_2: fixes x :: "'b set"   and xa :: "'b \<Rightarrow> 'c"   and xb :: "nat" assumes "higher_differentiable_on x xa xb" obtains S :: "'b set" and f :: "'b \<Rightarrow> 'c" where "x = S__" and "xa = f__" and "xb = 0" and "continuous_on S__ f__"   | S :: "'b set" and f :: "'b \<Rightarrow> 'c" and n :: "nat" where "x = S" and "xa = f" and "xb = Suc n" and "(\<forall>x\<in>S. f differentiable at x) \<and> (\<forall>v. higher_differentiable_on S (\<lambda>x. frechet_derivative f (at x) v) n)"
parts_image_Pan: fixes A :: "nat set" shows "parts (Pan ` A) = Pan ` A"
Compl_set_less_aux_Compl: fixes A :: "'a set"   and B :: "'a set" assumes "finite UNIV" shows "(- A \<sqsubset>' - B) = (B \<sqsubset>' A)"
double_add_le_zero_iff_single_add_le_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (plus a a) zero = less_eq a zero"
F_preserves_arrows: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" assumes "Functor F : AA \<longrightarrow> BB" shows "F \<^bsub>\<a>\<^esub> \<in> Ar\<^bsub>AA\<^esub> \<rightarrow> Ar\<^bsub>BB\<^esub>"
sqrt_nat_ceiling_code: fixes x :: "nat" shows "sqrt_nat_ceiling x = sqrt_int_ceiling_pos (int x)"
unit_disc_fix_f_unit_disc: fixes M :: "complex_homo \<Rightarrow> complex_homo" assumes "unit_disc_fix_f M" shows "M ` unit_disc = unit_disc"
A': shows "[:0::'a, 1::'a:] \<cdot>\<^sub>m 1\<^sub>m (dim_row A) + map_mat (\<lambda>a. [:- a:]) A \<in> carrier_mat n n"
simps: fixes \<tau>move :: "('c, 'd, 'a, 'b, 'e, 'f) \<tau>moves"   and a1 :: "(status \<times> 'a) \<times> 'b"   and a2 :: "('c, 'd, status \<times> 'a, 'b, 'e, 'f action) thread_action"   and a3 :: "(status \<times> 'a) \<times> 'b" shows "\<tau>multithreaded.init_fin_\<tau>move \<tau>move a1 a2 a3 = (\<exists>x m ta x' m'. a1 = ((Running, x), m) \<and> a2 = convert_TA_initial (convert_obs_initial ta) \<and> a3 = ((Running, x'), m') \<and> \<tau>move (x, m) ta (x', m'))"
cases: 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 a1 :: "'heap"   and a2 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and a3 :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and a4 :: "'heap"   and a5 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t a1 a2 a3 a4 a5" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and xcp :: "'addr option" and h' :: "'heap" and stk' :: "'addr val list" and loc' :: "'addr val list" and pc' :: "nat" and pc :: "nat" and h :: "'heap" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = ha__" and "a2 = (stka__, loca__, pca__, None)" and "a3 = ta" and "a4 = h'" and "a5 = (stk', loc', pc'a__, xcp__)" and "(ta, xcp__, h', [(stk', loc', undefined, undefined, pc'a__)]) \<in> exec_instr (ins ! pca__) P t ha__ stka__ loca__ undefined undefined pca__ []" and "pca__ < length ins" and "ci_app ci (ins ! pca__) P ha__ stka__ loca__ undefined undefined pca__ []"   | h :: "'heap" and xcp :: "'addr" and pc :: "nat" and pc' :: "nat" and d :: "nat" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = h" and "a2 = (stk, loc, pc, \<lfloor>xcp\<rfloor>)" and "a3 = \<lbrace>\<rbrace>" and "a4 = h" and "a5 = (Addr xcp # drop (length stk - d) stk, loc, pc', None)" and "match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor>" and "d \<le> length stk"
OK_lift2_OK: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b err"   and y :: "'d" shows "OK x \<squnion>\<^bsub>lift2 f\<^esub> OK y = x \<squnion>\<^bsub>f\<^esub> y"
impl_red_dfs_hash_def: shows "impl_red_dfs_hash \<equiv> \<lambda>x xa xb xc. REC\<^sub>T (\<lambda>xd (xe, xf). let y' = hs.ins xf xe in NDFS_SI_Statistics.vis_red_nres \<bind> (\<lambda>xg. LIST_FOREACH' (RETURN (id (x xf))) is_None (\<lambda>xh xi. if hs.memb xh xa then RETURN (red_init_witness xf xh) else RETURN None) None \<bind> (\<lambda>xh. case xh of None \<Rightarrow> LIST_FOREACH' (RETURN (id (x xf))) (\<lambda>(xi, y). is_None y) (\<lambda>xi (xj, xk). if \<not> hs.memb xi xj then xd (xj, xi) \<bind> (\<lambda>(xl, xm). RETURN (xl, prep_wit_red xf xm)) else RETURN (xj, None)) (y', None) | Some xi \<Rightarrow> RETURN (y', xh)))) (xb, xc)"
filter_True: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. P x" shows "filter P xs = xs"
dg_1_is_arrD_2: fixes \<aa> :: "V"   and \<ff> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_1 \<aa> \<ff>\<^esub> b" shows "b = \<aa>"
comp_mkArr: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and img :: "'a \<Rightarrow> 'a"   and \<AA> :: "('b \<times> 'b) set"   and A :: "'a set"   and B :: "'a set"   and F :: "'a \<Rightarrow> 'a"   and C :: "'a set"   and G :: "'a \<Rightarrow> 'a" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> B C G)" shows "S (set_category_given_img.mkArr S img \<AA> B C G) (set_category_given_img.mkArr S img \<AA> A B F) = set_category_given_img.mkArr S img \<AA> A C (G \<circ> F)"
fags_bppTr: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "fags_cond f z i a" shows "fags_bpp f z i a (n\<Odot>a\<^bsub>f,i,z\<^esub>) (m\<Odot>a\<^bsub>f,i,z\<^esub>) = (n + m)\<Odot>a\<^bsub>f,i,z\<^esub>"
vector_space_on_axioms: fixes p :: "'a" shows "vector_space_on (src.tangent_space p) (*\<^sub>R)"
connected_ball: fixes x :: "'b"   and e :: "real" shows "connected (ball x e)"
inverse_aform'_float: fixes p :: "nat"   and x :: "real \<times> real pdevs"   and a :: "real"   and bb :: "real pdevs"   and baa :: "real" assumes "inverse_aform' p x = ((a, bb), baa)" shows "a \<in> float"
ws_unique_fields_lemma: fixes efn :: "fspec"   and fd :: "Decl.field"   and G :: "prog"   and c :: "class"   and fc :: "fdecl"   and C :: "qtname"   and d :: "class"   and R :: "bool" assumes "(efn, fd) \<in> set (DeclConcepts.fields G (super c))"   and "fc \<in> set (cfields c)"   and "ws_prog G"   and "fname efn = fname fc"   and "declclassf efn = C"   and "class G C = Some c"   and "C \<noteq> Object"   and "class G (super c) = Some d" shows "R"
B: fixes x :: "'b" assumes "x \<in> (\<lambda>y. integral (cbox (fst y) (snd y)) f) ` cbox (a, a) (b, b)" shows "norm x \<le> B"
tllist_ord_ttlI: fixes b :: "'f"   and xs :: "('g, 'f) tllist"   and ys :: "('g, 'f) tllist" assumes "tllist_ord b xs ys" shows "tllist_ord b (ttl xs) (ttl ys)"
agree_comm: fixes A :: "(real, 'd) vec \<times> (real, 'd) vec"   and B :: "(real, 'd) vec \<times> (real, 'd) vec"   and V :: "('d + 'd) set" assumes "Vagree A B V" shows "Vagree B A V"
prems_1: shows "P,E \<turnstile> s \<surd>"
mru_of_set_def: fixes vs :: "'b \<Rightarrow> 'a \<Rightarrow> 'c option" shows "mru_of_set vs \<equiv> \<lambda>Q. option_Max_by fst (vote_set vs Q)"
adjacent: fixes w :: "'w"   and s :: "'w" assumes "w \<in> W"   and "s \<in> S" shows "dual_order.poset_simplex_map \<P> {w + s} \<sim> dual_order.poset_simplex_map \<P> {w}"
swap: fixes y :: "'b"   and x :: "'b"   and l :: "'b list" shows "y # x # l <~~> x # y # l"
exists_co_path_decomp1: fixes e :: "'a \<times> 'a"   and w :: "'a"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "e \<in> parcs G"   and "w \<notin> pverts G"   and "pre_digraph.apath (with_proj (subdivide G e w)) u p v"   and "(fst e, w) \<in> set p"   and "w \<noteq> v" shows "\<exists>p1 p2. p = p1 @ (fst e, w) # (w, snd e) # p2"
set_append: fixes xs :: "'e list"   and ys :: "'e list" shows "set (append xs ys) = set xs \<union> set ys"
inputTransition_def: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" shows "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<equiv> \<exists>P'''. P \<Longrightarrow>\<^sub>\<tau> P''' \<and> P''' \<longmapsto> a<x> \<prec> P'' \<and> P''[x::=u] \<Longrightarrow>\<^sub>\<tau> P'"
interchange: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a"   and \<sigma> :: "'a" assumes "B.seq \<nu> \<mu>"   and "B.seq \<tau> \<sigma>" shows "(\<nu> \<cdot>\<^sub>B \<mu>) \<star>\<^sub>B \<tau> \<cdot>\<^sub>B \<sigma> = (\<nu> \<star>\<^sub>B \<tau>) \<cdot>\<^sub>B \<mu> \<star>\<^sub>B \<sigma>"
\<tau>Runs_table_into_\<tau>Runs: fixes s :: "'s2"   and stlsss :: "('tl2 \<times> 's2, 's2 option) tllist" assumes "trsys2.\<tau>Runs_table s stlsss" shows "s \<Down>2 tmap fst id stlsss"
B: shows "ipurge_tr_rev_aux I D U (xs @ [x']) = ws @ [x]"
consistent_witness: fixes S :: "('a, 'b) fm set"   and p :: "('a, 'b) fm"   and used :: "'a set" assumes "consistent S"   and "p \<in> S"   and "params S \<subseteq> used"   and "finite used"   and "infinite UNIV" shows "consistent (witness used p \<union> S)"
elem_set: fixes x :: "'f"   and xo :: "'f option" shows "(x \<in> set_option xo) = (xo = Some x)"
strips_equivalent_to_sas_plus_i_a_II_2: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops :: "('c, 'd) sas_plus_operator list"   and op' :: "('c \<times> 'd) strips_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "op' \<in> set (map (sasp_op_to_strips \<Psi>) ops)" shows "is_valid_operator_strips (\<phi> \<Psi> ) op'"
sum_wt_ParT_sub_notWtFT_notFinished: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set"   and s :: "'state" assumes "properL cl"   and "WtFT cl \<noteq> 1"   and "n < length cl"   and "\<not> finished (cl ! n)"   and "I \<subseteq> {..<brn (cl ! n)}" shows "sum (wt (ParT cl) s) ((+) (brnL cl n) ` I) = 1 / real (length cl) / (1 - WtFT cl) * sum (wt (cl ! n) s) I"
cdqm_2: shows "p = Fun.swap c d id \<circ> q"
rank_monotonic: fixes x :: "nat"   and n :: "nat"   and i :: "nat"   and m :: "nat"   and j :: "nat" assumes "rank x n = Some i"   and "rank x (n + m) = Some j" shows "j \<le> i"
transform_simple_fw_lower_2: fixes rs :: "'c common_primitive rule list" assumes "simple_ruleset rs" shows "_Collect p (simple_fw (to_simple_firewall (lower_closure (optimize_matches abstract_for_simple_firewall (lower_closure (packet_assume_new rs))))) p = Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New) \<subseteq> _Collect p ((common_matcher, in_doubt_deny),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New)"
nth_list_of_vec: fixes i :: "nat"   and v :: "'a vec" assumes "i < dim_vec v" shows "list_of_vec v ! i = v $ i"
eq: fixes f :: "'a set \<Rightarrow> 'b"   and Q :: "'b \<Rightarrow> bool" shows "{y \<in> f ` \<A>. Q y} = f ` {x \<in> \<A>. Q (f x)}"
check_ipresIGFreshAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGFreshAll check check (errMOD MOD) MOD"
post_meas0_index_0_alice_state: fixes \<phi> :: "complex Matrix.mat" assumes "state 1 \<phi>" shows "state 3 (post_meas0 3 (alice \<phi>) 0)"
ninv_hom: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "f \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"   and "inj_on f {i. i \<le> n}" shows "ninv n f \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"
poly_even_odd: fixes f :: "'a poly"   and g :: "'a poly"   and h :: "'a poly" assumes "poly_even_odd f = (g, h)" shows "f = g \<circ>\<^sub>p Polynomial.monom (1::'a) 2 + Polynomial.monom (1::'a) 1 * h \<circ>\<^sub>p Polynomial.monom (1::'a) 2"
atom_map_compl_pres: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot (uminus x) = Collect (bounded_lattice.atom less bot) - bounded_lattice.atom_map less_eq less bot x"
assms_2: shows "(qinp, qinp') \<in> alphaInp"
MakeFormP_graph_aux_def: shows "MakeFormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>MakeFormP_aux v y u w au. x1 = (y, u, w) \<and> x2 = y EQ Q_Disj u w OR y EQ Q_Neg u OR SyntaxN.Ex v (SyntaxN.Ex au (AbstFormP (Var v) Zero u (Var au) AND y EQ Q_Ex (Var au))) \<and> atom v \<sharp> (y, u, w, au) \<and> atom au \<sharp> (y, u, w))"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelRTPO Enc TRel"   and "(Q, R) \<in> encoding.indRelRTPO Enc TRel" shows "(P, R) \<in> encoding.indRelRTPO Enc TRel"
locally_path_connected_space_def: fixes X :: "'a topology" shows "locally_path_connected_space X \<equiv> neighbourhood_base_of (path_connectedin X) X"