hcju/isabelleps huggingface dataset:
rel_mono_strong: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a set['c]"   and y :: "'b set['c]"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_bset R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_bset x; yb \<in> set_bset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_bset Ra x y"
not_MTimpnotMT: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and p :: "'a list" assumes "not_MT C p" shows "p \<noteq> []"
simply_connected_eq_biholomorphic_to_disc: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (S = {} \<or> S = UNIV \<or> (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and> (\<forall>z\<in>S. f z \<in> ball 0 1 \<and> g (f z) = z) \<and> (\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z)))"
collinear_3_affine_hull: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<noteq> b" shows "collinear {a, b, c} = (c \<in> affine hull {a, b})"
preserves_arr: fixes f :: "'b \<times> 'b" assumes "B.VV.arr f" shows "B.hseq (fst f) (snd f)"
monom_mult_list_inv: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m1 :: "('c \<times> nat) list"   and m2 :: "('c \<times> nat) list" assumes "class.linorder less_eq less"   and "linorder.monom_inv less_eq m1"   and "linorder.monom_inv less_eq m2" shows "linorder.monom_inv less_eq (linorder.monom_mult_list less m1 m2)"
assms_1: shows "(x, y) \<in> E'\<^sup>*"
finprod_def: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" shows "finprod G f A = (if finite A then foldD (carrier G) ((\<otimes>\<^bsub>G\<^esub>) \<circ> f) \<one>\<^bsub>G\<^esub> A else \<one>\<^bsub>G\<^esub>)"
lcf_val: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "lcf R S X p = snd (s_cf R S X p) (fst (s_cf R S X p))"
nsZero: fixes n :: "nat" shows " n \<times>\<^bsub>R\<^esub> \<zero> = \<zero>"
setge_def: fixes x :: "'a"   and S :: "'a set" shows "x <=* S = (\<forall>y\<in>S. x \<le> y)"
deduct_num_if_deduct: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "M \<turnstile> t" shows "\<exists>k. \<langle>M; k\<rangle> \<turnstile>\<^sub>n t"
P1_ideal_ideal_eq: fixes x :: "'grp \<times> 'grp"   and y :: "bool"   and z :: "'e"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)" shows "mal_def.ideal_view_1 x y z (P1_S1, P1_S2) \<A> = P1_ideal_model x y z \<A>"
get_child_nodes_ok: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr"   and "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr"
TS_finite: fixes V :: "'d list set"   and M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and m :: "nat"   and n :: "nat" assumes "finite V"   and "finite (inputs M2)" shows "finite (TS M2 M1 \<Omega> V m n)"
map_iteratei_def: fixes \<alpha> :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and invar :: "'c \<Rightarrow> bool"   and iteratei :: "'c \<Rightarrow> ('f \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f" shows "map_iteratei \<alpha> invar iteratei \<equiv> finite_map \<alpha> invar \<and> map_iteratei_axioms \<alpha> invar iteratei"
prems_2: shows "A $ i $ mod_type_class.from_nat k \<noteq> (0::'a)"
image_add_atLeastLessThan': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "'b"   and i :: "'b"   and j :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "(\<lambda>n. plus n k) ` ord.atLeastLessThan less_eq less i j = ord.atLeastLessThan less_eq less (plus i k) (plus j k)"
finite_measure_mono_AE: fixes A :: "'a set"   and B :: "'a set" assumes "AE x\<in>A in M. x \<in> B"   and "B \<in> sets M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M B"
strict_sorted_sorted_wrt: shows "local.strict_sorted = sorted_wrt lt"
exhaust: fixes y :: "'x list" obtains "y = []"   | x21 :: "'x" and x22 :: "'x list" where "y = x21 # x22"
exhaust: fixes y :: "('b, 'c) state" obtains x :: "'b \<Rightarrow> 'c \<times> 'b" where "y = State x"
extend_map_UNIV_to_sphere_no_bounded_component: fixes r :: "real"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b" assumes "DIM('a) \<le> DIM('b)"   and "0 \<le> r"   and "compact S"   and "continuous_on S f"   and "f ` S \<subseteq> sphere a r"   and "\<And>C. C \<in> components (- S) \<Longrightarrow> \<not> bounded C" obtains g :: "'a \<Rightarrow> 'b" where "continuous_on UNIV g" and "range g \<subseteq> sphere a r" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
while_eval': fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>While e c, mds, mem\<rangle> \<leadsto> \<langle>Stmt.If e (c ;; While e c) Stop, mds, mem\<rangle>"
Tree_wf\<^sub>pI: fixes a :: "('e, 'f, 'g) Tree"   and b :: "('e, 'f, 'g) Tree"   and p :: "perm" assumes "(a, b) \<in> Tree_wf" shows "(abs_Tree\<^sub>p (p \<bullet> a), abs_Tree\<^sub>p b) \<in> Tree_wf\<^sub>p"
preserves_seq: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" shows "D.seq (D.R g) (D.R f)"
Obj_implies_Ide: fixes t :: "'c local.term" assumes "Obj t" shows "Ide t"
isoI: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "h \<in> hom G H"   and "bij_betw h (carrier G) (carrier H)" shows "h \<in> Group.iso G H"
gen_step_def: fixes s :: "'s" shows "gen_step s \<equiv> if gds_is_empty_stack gds s then SPEC (\<lambda>v0. v0 \<in> V0 \<and> \<not> gds_is_discovered gds v0 s) \<bind> (\<lambda>v0. gds_new_root gds v0 s) else gds_get_pending gds s \<bind> (\<lambda>(u, Vs, s). case Vs of None \<Rightarrow> gds_finish gds u s | Some v \<Rightarrow> if gds_is_discovered gds v s then if gds_is_finished gds v s then gds_cross_edge gds u v s else gds_back_edge gds u v s else gds_discover gds u v s)"
simps_3: fixes m :: "nat"   and n :: "nat" shows "arity (recf.Id m n) = m"
emb_step_hsize: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "hsize s < hsize t"
dg_10_is_arr_iff: fixes \<aa> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<^bsub>dg_10 \<aa>\<^esub> \<BB> = False"
hiding_advantage_ind_cpa_def: fixes \<A> :: "('pub_input \<times> 'witness \<Rightarrow> (('challenge \<times> 'challenge) \<times> 'b) spmf) \<times> ('msg \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "abstract_com.hiding_advantage_ind_cpa \<A> \<equiv> \<bar>spmf (abstract_com.hiding_game_ind_cpa \<A>) True - 1 / 2\<bar>"
Sigma: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "(\<Sum>x\<in>A. sum (g x) (B x)) = (\<Sum>(x, y)\<in>Sigma A B. g x y)"
fn_norm_leastB: fixes f :: "'a \<Rightarrow> real"   and y :: "real" assumes "Function_Norm.continuous V f norm"   and "\<And>b. b \<in> B V f \<Longrightarrow> b \<le> y" shows "\<parallel>f\<parallel>\<hyphen>V \<le> y"
elims: fixes x :: "int list"   and y :: "int list" assumes "take_decr x = y" obtains "x = []" and "y = []"   | xa :: "int" where "x = [xa__]" and "y = [xa__]"   | xa :: "int" and ya :: "int" and xs :: "int list" where "x = xa # ya # xs" and "y = (if ya \<le> xa then xa # take_decr (ya # xs) else [xa])"
design_blocks_nempty: shows "arcs_blocks \<noteq> {#}"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "(case ERR of ERR \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = f1"
N_2: fixes y :: "'a" assumes "y \<in> X" shows "{x \<in> space M. \<not> P x y} \<subseteq> N y"
empty: fixes g :: "'b \<Rightarrow> 'a" shows "sum g {} = (0::'a)"
normalize_unit_factor_eqI: fixes a :: "'a"   and b :: "'a" assumes "normalize a = normalize b"   and "unit_factor a = unit_factor b" shows "a = b"
quotient_is_ring: shows "ring (poly_ring L Quot PIdl\<^bsub>poly_ring L\<^esub> q)"
mult_strict_right_mono_neg: 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less b a"   and "less c zero" shows "less (times a c) (times b c)"
encL: fixes S :: "'procS" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L SourceTerm S"
sm_QF1: fixes i :: "nat"   and ka :: "nat"   and k :: "nat" assumes "i < m"   and "ka \<le> Suc k"   and "k \<le> ka" shows "strict_mono_sets {..<ka} (\<lambda>j. QF k (j, i))"
cases: fixes x :: "'a list list" obtains "x = []"   | v :: "'a list" and va :: "'a list list" where "x = v # va"
SEQ_iff: fixes f :: "nat \<Rightarrow> 'b"   and A :: "'b set" shows "(f \<in> SEQ A) = (\<forall>i. f i \<in> A)"
order_trans_classes1: fixes c :: "classes1"   and d :: "classes1"   and e :: "classes1" assumes "c \<le> d"   and "d \<le> e" shows "c \<le> e"
lowner_lub_add: fixes d :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and g :: "nat \<Rightarrow> complex mat" assumes "matrix_seq d f"   and "matrix_seq d g"   and "\<forall>n. trace (f n + g n) \<le> 1" shows "matrix_seq.lowner_lub (\<lambda>n. f n + g n) = matrix_seq.lowner_lub f + matrix_seq.lowner_lub g"
handleE'_def: fixes f :: "'c \<Rightarrow> (('f + 'e) \<times> 'c) set \<times> bool"   and handler :: "'f \<Rightarrow> 'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool" shows "f <handle2> handler \<equiv> do v \<leftarrow> f; case v of Inl e \<Rightarrow> handler e | Inr v' \<Rightarrow> return (Inr v') od"
absorb_iff2: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (sup a b = b)"
R: shows "\<forall>a b. negligible (s \<inter> cbox a b)"
interpret_floatarith_FDERIV_floatarith_eq: fixes xs :: "nat list"   and ds :: "floatarith list"   and fa :: "floatarith"   and vs :: "real list" assumes "length xs = DIM('a)"   and "length ds = DIM('a)" shows "interpret_floatarith (FDERIV_floatarith fa xs ds) vs = einterpret (map (\<lambda>x. DERIV_floatarith x fa) xs) vs \<bullet> einterpret ds vs"
simps_2: shows "prev (Suc 0) 0 = None"
ring_id_eval: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "ring_id a = a"
cases: fixes t :: "'t"   and a1 :: "'x1 \<times> 'm1"   and a2 :: "'x1 \<times> 'm1" assumes "r1.silent_move t a1 a2" obtains s :: "'x1 \<times> 'm1" and s' :: "'x1 \<times> 'm1" and tl :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action" where "a1 = s" and "a2 = s'" and "t \<turnstile> s -1-tl\<rightarrow> s'" and "\<tau>move1 s tl s'"
ex_finite_subset_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and P :: "'c set \<Rightarrow> bool" shows "(\<exists>B. finite B \<and> B \<subseteq> f ` A \<and> P B) = (\<exists>B. finite B \<and> B \<subseteq> A \<and> P (f ` B))"
upper_remainder: fixes Cn :: "'b set \<Rightarrow> 'b set"   and B :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "Compact_logic Cn"   and "B \<subseteq> A"   and "\<phi> \<notin> Cn B" shows "\<exists>B'. B \<subseteq> B' \<and> B' \<in> Tarskian_logic.remainder Cn A \<phi>"
assoc_mult_mat_vec: fixes A :: "'c mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'c mat"   and n\<^sub>3 :: "nat"   and v :: "'c vec" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "v \<in> carrier_vec n\<^sub>3" shows "A * B *\<^sub>v v = A *\<^sub>v (B *\<^sub>v v)"
transpose_aux_filter_head: fixes xss :: "'g list list" shows "concat (map (case_list [] (\<lambda>h t. [h])) xss) = map hd (filter (\<lambda>ys. ys \<noteq> []) xss)"
carrier_update_4: fixes S :: "'a set" shows "(\<oplus>\<^bsub>R\<lparr>carrier := S\<rparr>\<^esub>) = (\<oplus>)"
order_iff_strict: fixes a :: "'a word"   and b :: "'a word" shows "(a \<le>s b) = (a <s b \<or> a = b)"
Max_wrt_subset: shows "Max_wrt R \<subseteq> alts"
fls_divide_nth_base: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_nth (f / g) (fls_subdegree f - fls_subdegree g) = fls_nth f (fls_subdegree f) / fls_nth g (fls_subdegree g)"
distrib: fixes g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "F.F (\<lambda>x. g x \<^bold>* h x) A = F.F g A \<^bold>* F.F h A"
nv2L_Wax: fixes c' :: "(('a, 'tp) efsym, ('psym, 'tp) epsym) lit list"   and l' :: "(('a, 'tp) efsym, ('psym, 'tp) epsym) lit" assumes "c' \<in> Wax"   and "l' \<in> set c'" shows "nv2L l' = {}"
mult_div_mod_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times b (divide a b)) (modulo a b) = a"
is_rbt_fold_rbt_insertwk: fixes t1 :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t2 :: "('a, 'c) rbt" assumes "is_rbt t1" shows "is_rbt (RBT_Impl.fold (rbt_insert_with_key f) t2 t1)"
Ints_mult: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus"   and "b \<in> ring_1.Ints one plus zero minus" shows "times a b \<in> ring_1.Ints one plus zero minus"
assms_3: fixes y :: "'a set" assumes "SimplicialComplex.maxsimp Y y" shows "chamber y"
is_category: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "product_category C1 C2" shows "category (product_category.comp C1 C2)"
l2_when_right_child: fixes l :: "nat"   and l' :: "nat"   and i :: "int"   and i' :: "int" assumes "l < l'"   and "i * 2 ^ (l' - l) < i'"   and "i' < (i + 1) * 2 ^ (l' - l)" shows "l2_\<phi> (l', i') (l, i) = (1 - real_of_int i' / 2 ^ (l' - l) + real_of_int i) / 2 ^ (l' + 1)"
lr_of_tran_s1_def: fixes rt :: "('c, 'd) routing_rule_scheme list" shows "lr_of_tran_s1 rt = map (\<lambda>r. (route2match r, routing_oiface r)) rt"
ge_iff_diff_ge_0: fixes b :: "'a"   and a :: "'a" shows "b \<le> a \<equiv> (0::'a) \<le> a - b"
asset_ii_used: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "(Asset n, \<lbrace>Num 2, PubKey A\<rbrace>) \<in> s \<longrightarrow> PriKey A \<in> used s"
step2: shows "length M2 \<le> j \<longrightarrow> (i < row_length (v # M) * row_length M2 \<and> j < length (v # M) * length M2) \<and> mat (row_length (v # M)) (length (v # M)) (v # M) \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> (v # M \<otimes> M2) ! j ! i = (v # M) ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
sturm_squarefree'_signs: fixes p :: "real poly"   and i :: "nat"   and x :: "real" assumes "p \<noteq> 0"   and "i < length (sturm_squarefree' p) - 2"   and "poly (sturm_squarefree' p ! (i + 1)) x = 0" shows "poly (sturm_squarefree' p ! (i + 2)) x * poly (sturm_squarefree' p ! i) x < 0"
1: shows "t + punit.lt (rep_list f) \<in> keys (rep_list p)"
incomplete_designI: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size"   and "u_block_size < int (card point_set)" shows "incomplete_design point_set block_collection u_block_size"
test_def: fixes f :: "real set \<Rightarrow> real set" shows "test f \<equiv> map (\<lambda>x. x \<in> f RRR) [1, 2, 3, 4, 6]"
assms_4: shows "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>"
VDM_valid_validn: fixes c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and n :: "nat" assumes " \<Turnstile> c : A " shows " \<Turnstile>\<^sub>n c : A "
timpl_closure_subst: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and \<delta> :: "(('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat \<Rightarrow> (('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and T :: "('sets set \<times> 'sets set) set" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<forall>x\<in>fv t. \<exists>a. \<Gamma>\<^sub>v x = Var (prot_atom.Atom a)"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "timpl_closure (t \<cdot> \<delta>) T = timpl_closure t T \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
Pi_split_insert_domain: fixes x :: "'d \<Rightarrow> 'e"   and i :: "'d"   and I :: "'d set"   and X :: "'d \<Rightarrow> 'e set" shows "(x \<in> Pi (insert i I) X) = (x \<in> Pi I X \<and> x i \<in> X i)"
not_Suc_Greatest_not_Suc: fixes P :: "nat \<Rightarrow> bool"   and x :: "nat" assumes "\<not> P (Suc x)"   and "\<exists>x. P x" shows "(GREATEST y. y \<le> x \<and> P y) = (GREATEST y. y \<le> Suc x \<and> P y)"
antisym: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "ordering less_eq less"   and "less_eq a b"   and "less_eq b a" shows "a = b"
sum_atLeastAtMost_int_head: fixes a :: "int"   and b :: "int"   and f :: "int \<Rightarrow> 'a" assumes "a < b" shows "sum f {a..b} = f a + sum f {a<..b}"
isSome_simps_2: fixes x :: "'b option" shows "(\<not> isSome x) = (x = None)"
fix_eq3: fixes f :: "'b \<rightarrow> 'c"   and F :: "('b \<rightarrow> 'c) \<rightarrow> 'b \<rightarrow> 'c"   and x :: "'b" assumes "f \<equiv> fix\<cdot>F" shows "f\<cdot>x = F\<cdot>f\<cdot>x"
prime_factors_dvd: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "prime_factors x = {p. prime p \<and> p dvd x}"
diff_card_le_card_Diff: fixes B :: "'b set"   and A :: "'b set" assumes "finite B" shows "card A - card B \<le> card (A - B)"
lc_zero: shows "punit.lc 0 = (0::'c)"
eval_extract_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>'c\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and f :: "nat \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'b" shows "eval_extract X f = (\<lambda>\<tau>. case X \<tau> of \<bottom> \<Rightarrow> invalid \<tau> | \<lfloor>\<bottom>\<rfloor> \<Rightarrow> invalid \<tau> | \<lfloor>\<lfloor>obj\<rfloor>\<rfloor> \<Rightarrow> f (oid_of obj) \<tau>)"
adj_not_same: fixes G :: "('b, 'c) pre_digraph"   and a :: "'b" assumes "loopfree_digraph G"   and "a \<rightarrow>\<^bsub>G\<^esub> a" shows "False"
hom_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
v_def: fixes f :: "'a \<Rightarrow> real" shows "v f = w (extend f)"
del_min_def: fixes ts :: "'b tree list" shows "del_min ts = (case get_min_rest ts of (Node r x ts\<^sub>1, ts\<^sub>2) \<Rightarrow> merge (rev ts\<^sub>1) ts\<^sub>2)"
S: fixes x :: "nat"   and i :: "nat" shows "(x \<in> succeed_t i) = (r x \<in> succeed\<^sub>R i)"
has_domain_char: fixes f :: "'a \<times> 'a" shows "(CC.domains f \<noteq> {}) = (arr (fst f) \<and> arr (snd f))"
index_mat_four_block_1: fixes i :: "nat"   and A :: "'a mat"   and D :: "'a mat"   and j :: "nat"   and B :: "'a mat"   and C :: "'a mat" assumes "i < dim_row A + dim_row D"   and "j < dim_col A + dim_col D" shows "four_block_mat A B C D $$ (i, j) = (if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A) else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))"
test_mult_idem_var: fixes p :: "'a" assumes "test p" shows "p \<cdot> p = p"
vfsequence_vcons_vproductD_2: fixes xs :: "V"   and x :: "V"   and n :: "V"   and A :: "V \<Rightarrow> V" assumes "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> vproduct n A"   and "n \<in>\<^sub>\<circ> \<omega>" shows "x \<in>\<^sub>\<circ> A (vcard xs)"
stutter_invariantD: fixes \<phi> :: "'b pltl"   and \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "stutter_invariant \<phi>"   and "\<sigma> \<approx> \<tau>" shows "\<sigma> \<Turnstile>\<^sub>p \<phi> = \<tau> \<Turnstile>\<^sub>p \<phi>"
short: shows "\<^bold>|x\<^bold>| + \<^bold>|x\<^bold>| \<le> \<^bold>|w\<^bold>|"
simps_3: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs \<or> \<not> lnull ys" shows "lhd (lappend xs ys) = (case xs of LNil \<Rightarrow> lhd ys | LCons x xs' \<Rightarrow> x)"
PTrans: shows "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
ffact_Suc: fixes n :: "nat"   and a :: "'b" shows "ffact (Suc n) a = a * ffact n (a - (1::'b))"
complete_algebra_summable_geometric: fixes x :: "'a" assumes "norm x < 1" shows "summable ((^) x)"
verts_del_vert: fixes u :: "'a" shows "verts (H.del_vert u) = verts (del_vert u) - {u}"
length_subseqs: fixes xs :: "'b list" shows "length (subseqs xs) = 2 ^ length xs"
Ctxt_valid_validn: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and n :: "nat" assumes " \<Turnstile> G " shows " \<Turnstile>\<^sub>n G  "
LN_4: fixes \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> lfilledk : ts _> ts'" shows "\<exists>t1s t2s \<C>' arb_label. \<S>\<bullet>\<C>\<lparr>label := arb_label @ label \<C>\<rparr> \<turnstile> es : t1s _> t2s"
is_document_ptr_kind_none: fixes ptr :: "(_) object_ptr" assumes "\<not> is_document_ptr_kind ptr" shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = None"
poly_mult_in_carrier: fixes R :: "('c, 'd) ring_scheme"   and p1 :: "'c list"   and p2 :: "'c list" assumes "ring R"   and "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "set (ring.poly_mult R p1 p2) \<subseteq> carrier R"
fps_sum_rep_nth: fixes a :: "'a fps"   and m :: "nat"   and n :: "nat" shows "fps_nth (\<Sum>i = 0..m. fps_const (fps_nth a i) * fps_X ^ i) n = (if n \<le> m then fps_nth a n else (0::'a))"
conseq: fixes P :: "'b assn"   and G :: "prog"   and A :: "'b triple set"   and t :: "Term.term"   and Q :: "'b assn" assumes "\<forall>Y s Z. P Y s Z \<longrightarrow> (\<exists>P' Q'. G,A\<turnstile>{P'} t\<succ> {Q'} \<and> (\<forall>Y' s'. (\<forall>Y Z'. P' Y s Z' \<longrightarrow> Q' Y' s' Z') \<longrightarrow> Q Y' s' Z))" shows "G,A\<turnstile>{P} t\<succ> {Q}"