hcju/isabelleps huggingface dataset:
spec2_assoc: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point"   and p\<^sub>3 :: "'b point" assumes "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "is_generic p\<^sub>1 p\<^sub>2"   and "is_tangent p\<^sub>2 p\<^sub>3"   and "is_generic (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"   and "is_generic p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3)" shows "ell_field_class.add a p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3) = ell_field_class.add a (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
isomorphic_homology_imp_path_components: fixes X :: "'b topology"   and Y :: "'c topology" assumes "homology_group 0 X \<cong> homology_group 0 Y" shows "path_components_of X \<approx> path_components_of Y"
intro: fixes Obj :: "'a set"   and Arr :: "'b set"   and D :: "'b \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'a" assumes "graph Obj Arr D C" shows "free_category Obj Arr D C"
berlekamp_monic_factorization_def: fixes d :: "nat"   and f :: "'a mod_ring poly" shows "berlekamp_monic_factorization d f = (let vs = berlekamp_basis f; n = length vs; fs = berlekamp_factorization_main d [f] vs n in fs)"
prod_BallI: fixes M :: "('a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. (a, b) \<in> M \<Longrightarrow> P a b" shows "\<forall>(a, b)\<in>M. P a b"
char_eq_0_iff: fixes G :: "('a, 'b) monoid_scheme"   and \<chi> :: "'a \<Rightarrow> complex"   and a :: "'a" assumes "character G \<chi>" shows "(\<chi> a = 0) = (a \<notin> carrier G)"
n_intersect_num_subset_def: fixes b1 :: "'b set"   and n :: "nat"   and b2 :: "'b set" shows "(b1 |\<inter>|\<^sub>n b2) = int (card {x. x \<subseteq> b1 \<inter> b2 \<and> card x = n})"
F_ctor_o_rec: fixes s :: "('e, 'e F \<times> 'f) F0 \<Rightarrow> 'f" shows "ctor_rec_F s \<circ> ctor_F = s \<circ> map_pre_F id <id, ctor_rec_F s>"
fs2: shows "c' \<sharp> (M, M', P, P, c)"
A2: shows "\<forall>u. c_fold (c_unfold k u) = u"
semiring_0_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "class.semiring_0 plus zero times \<equiv> class.comm_monoid_add plus zero \<and> class.mult_zero times zero \<and> class.semiring plus times"
simps: fixes M :: "('e, 'f, 'd) FSM"   and p :: "'d" shows "d_reachable M p = {q. \<exists>xs. d_reaches M p xs q}"
zero_real_def: shows "0 \<equiv> Real.Real (\<lambda>n. 0)"
fst_snd_A': shows "fst (snd A') = fst (snd B)"
vpair_def': fixes a :: "V"   and b :: "V" shows "\<langle>a, b\<rangle> = set {set {a, a}, set {a, b}}"
mk_finally_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (mk_finally \<phi>) \<subseteq> atoms_ltln \<phi>"
sets_lhs: shows "sets (M \<bind> (\<lambda>x. if P x then g x else h x)) = sets R"
sorted_wrt1: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" shows "sorted_wrt P [x] = True"
even_or_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (or a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<and> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
n_le_fact: fixes n :: "nat" shows "n < Suc (n!)"
arr_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" shows "S.arr (S.mkArr A B F) = (A \<subseteq> S.Univ \<and> B \<subseteq> S.Univ \<and> F \<in> A \<rightarrow> B)"
simps_1: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree" shows "BinomialHeapStruc.ins t [] = [t]"
intros_20: fixes \<C> :: "t_context"   and ts :: "t list"   and "is" :: "nat list"   and i :: "nat"   and t1s :: "t list"   and t2s :: "t list" assumes "list_all (\<lambda>i. i < length (label \<C>) \<and> label \<C> ! i = ts) (is @ [i])" shows "\<C> \<turnstile> [Br_table is i] : t1s @ ts @ [T_i32] _> t2s"
aAa: shows "adjoint A = adjoint (P * B * adjoint P)"
sums_scaleR_left: fixes X :: "nat \<Rightarrow> real"   and a :: "real"   and x :: "'b" assumes "X sums a" shows "(\<lambda>n. X n *\<^sub>R x) sums (a *\<^sub>R x)"
elims_3: fixes x :: "'a simple_match"   and xa :: "('a, 'b) simple_packet_scheme" assumes "\<not> simple_matches x xa" obtains m :: "'a simple_match" and p :: "('a, 'b) simple_packet_scheme" where "x = m" and "xa = p" and "\<not> (match_iface (iiface m) (p_iiface p) \<and> match_iface (oiface m) (p_oiface p) \<and> simple_match_ip (src m) (p_src p) \<and> simple_match_ip (dst m) (p_dst p) \<and> match_proto (proto m) (p_proto p) \<and> simple_match_port (sports m) (p_sport p) \<and> simple_match_port (dports m) (p_dport p))"
partition_on_msetD1: fixes A :: "'a multiset"   and P :: "'a multiset multiset" assumes "partition_on_mset A P" shows "\<Sum>\<^sub># P = A"
llist_of_list_of: fixes xs :: "'a llist" assumes "lfinite xs" shows "llist_of (list_of xs) = xs"
show_real_def: fixes x :: "real" shows "show_real x \<equiv> if \<exists>y. x = real_of_u y then mau_show_real (THE y. x = real_of_u y) else []"
s_inj_1: fixes m :: "nat"   and m' :: "nat" assumes "s1 m = s1 m'" shows "m = m'"
setcat_def: fixes \<AA> :: "('a \<times> 'a) set" shows "setcat \<AA> \<equiv> Card_order \<AA> \<and> infinite (Field \<AA>)"
PPf_def: shows "PPf \<equiv> \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)]"
dist_add_vec: fixes x :: "'b"   and s :: "real"   and v :: "'b" shows "dist (x + s *\<^sub>R v) x = \<bar>s\<bar> * norm v"
simps_8: fixes var :: "nat"   and \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "convert_fm var (AllQ \<phi>) \<Gamma> = None"
comp_assoc: fixes h :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" shows "C.VV.comp (C.VV.comp h g) f = C.VV.comp h (C.VV.comp g f)"
ordinal_of_hmset_iff: fixes k :: "hmultiset" shows "(ordinal_of_hmset k = 0) = (k = 0)"
Pos_Melem_subst_atm_subst_cls: fixes A :: "'a"   and C :: "'a literal multiset"   and \<sigma> :: "'s" assumes "Pos A \<in># C" shows "Pos (A \<cdot>a \<sigma>) \<in># C \<cdot> \<sigma>"
same_prefix_prefix: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" shows "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
mat_plus_comm: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and m1 :: "'c list list"   and m2 :: "'c list list" assumes "class.ab_semigroup_add plus" shows "mat_plusI plus m1 m2 = mat_plusI plus m2 m1"
measurable_distr_eq1: fixes Mf :: "'e measure"   and Nf :: "'c measure"   and f :: "'e \<Rightarrow> 'c"   and Mf' :: "'d measure" shows "distr Mf Nf f \<rightarrow>\<^sub>M Mf' = Nf \<rightarrow>\<^sub>M Mf'"
eq_fold: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and M :: "'a multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z M = fold_mset f z M"
aadd_assoc_i: fixes x :: "ant"   and y :: "ant"   and z :: "ant" assumes "x \<in> Z\<^sub>\<infinity>"   and "y \<in> Z\<^sub>\<infinity>"   and "z \<in> Z\<^sub>\<infinity>" shows "x + y + z = x + (y + z)"
ord_le_eq_trans: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a \<le>s b"   and "b = c" shows "a \<le>s c"
good_def: fixes xs :: "'a list" shows "good xs = hd xs__ \<leadsto>xs\<leadsto> last xs__"
multiple_t_wise_balance_index: fixes ps :: "'a set"   and n :: "nat" assumes "ps \<subseteq> \<V>"   and "int (card ps) = \<t>" shows "int (multiple_blocks n index ps) = \<Lambda>\<^sub>t * int n"
valid_SDG_node_in_slice_parent_node_in_slice: fixes n :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "n \<in> HRB_slice S" shows "CFG_node (parent_node n) \<in> HRB_slice S"
distinct_left: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree"   and y :: "'b" assumes "all_distinct (Node l x False r)"   and "y \<in> set_of l" shows "x \<noteq> y"
simps_1: fixes k :: "'b" shows "cnf (Atom k) = {{k\<^sup>+}}"
mult_product_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" shows "(\<otimes>\<^bsub>product_group I G\<^esub>) = (\<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i)"
conf_litval: fixes v :: "val"   and T :: "ty"   and G :: "prog"   and s :: "st" assumes "typeof Map.empty v = Some T" shows "G,s\<turnstile>v\<Colon>\<preceq>T"
hyps_3: shows "\<Gamma>\<turnstile> \<langle>c',s'\<rangle> \<Rightarrow> Abrupt t"
induct: fixes x1 :: "'a list"   and x2 :: "'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "x1 <~~> x2"   and "P [] []"   and "\<And>y x l. P (y # x # l) (x # y # l)"   and "\<And>xs ys z. \<lbrakk>xs <~~> ys; P xs ys\<rbrakk> \<Longrightarrow> P (z # xs) (z # ys)"   and "\<And>xs ys zs. \<lbrakk>xs <~~> ys; P xs ys; ys <~~> zs; P ys zs\<rbrakk> \<Longrightarrow> P xs zs" shows "P x1 x2"
inducts: fixes x :: "('state, 'act, 'out) trans list"   and P :: "('state, 'act, 'out) trans list \<Rightarrow> bool" assumes "valid2 x"   and "\<And>trn. validTrans trn \<Longrightarrow> P [trn]"   and "\<And>tr trn. \<lbrakk>valid2 tr; P tr; tgtOf (last tr) = srcOf trn; validTrans trn\<rbrakk> \<Longrightarrow> P (tr ## trn)" shows "P x"
inherits_def: fixes G :: "prog"   and C :: "qtname"   and m :: "qtname \<times> memberdecl" shows "G \<turnstile> C inherits m = (G \<turnstile> m inheritable_in pid C \<and> G\<turnstile> memberid m undeclared_in C \<and> (\<exists>S. G\<turnstile>C\<prec>\<^sub>C1S \<and> G \<turnstile> Class S accessible_in pid C \<and> G \<turnstile> m member_of S))"
simps: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.height_tree (SkewBinomialHeapStruc.Node e a r ts) = SkewBinomialHeapStruc.height_queue ts"
expand: fixes gpv :: "('b, 'd, 'e) gpv"   and gpv' :: "('b, 'd, 'e) gpv" assumes "the_gpv gpv = the_gpv gpv'" shows "gpv = gpv'"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b list_R1"   and b :: "'c list_R1" shows "rel_list_R1 R a b = rel_sum (=) (rel_prod R (rel_list_R1 R)) (unR a) (unR b)"
subgroup_kernel: shows "subgroup (kernel G (H\<lparr>carrier := h ` carrier G\<rparr>) h) G"
iMOD_cut_greater: fixes r :: "nat"   and m :: "nat"   and t :: "nat" shows "[ r, mod m ] \<down>> t = (if t < r then [ r, mod m ] else if m = 0 then {} else [ t + m - (t - r) mod m, mod m ])"
Morse_Gromov_theorem: fixes lambda :: "real"   and C :: "real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and G :: "'a set" assumes "lambda C -quasi_isometry_on {a..b} f"   and "geodesic_segment_between G (f a) (f b)" shows "hausdorff_distance (f ` {a..b}) G \<le> 92 * lambda\<^sup>2 * (C + deltaG TYPE('a))"
ds_def: shows "ds = Abs_downset \<circ> \<down>"
value_k_i: fixes i :: "nat" assumes "flow_network \<Delta> (f_i i)"   and "value_flow \<Delta> (f_i i) = value_flow \<Delta> (f_aux i)" shows "value_flow (residual_network (f_i i)) (k_i i) = value_flow (residual_network (f_i i)) (f_aux (Suc i) \<ominus> f_i i)"
get_dom_component_parent_inside: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and node_ptr :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \<in> set c"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r Some parent" shows "parent \<in> set c"
cases: fixes a1 :: "nat"   and a2 :: "com" assumes "a1 \<turnstile> a2" obtains l :: "nat" where "a1 = la__" and "a2 = SKIP"   | x :: "char list" and a :: "aexp" and l :: "nat" where "a1 = lb__" and "a2 = x ::= a" and "sec a \<le> sec x" and "lb__ \<le> sec x"   | l :: "nat" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "a1 = lc__" and "a2 = c\<^sub>1__;; c\<^sub>2__" and "lc__ \<turnstile> c\<^sub>1__" and "lc__ \<turnstile> c\<^sub>2__"   | b :: "bexp" and l :: "nat" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "a1 = ld__" and "a2 = IF b__ THEN c\<^sub>1 ELSE c\<^sub>2" and "max (sec b__) ld__ \<turnstile> c\<^sub>1" and "max (sec b__) ld__ \<turnstile> c\<^sub>2"   | b :: "bexp" and l :: "nat" and c :: "com" where "a1 = l" and "a2 = WHILE b DO c" and "max (sec b) l \<turnstile> c"
simps_2: fixes k :: "'a"   and k' :: "'a"   and v :: "'b"   and xs :: "('a \<times> 'b) list" shows "AList.delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # AList.delete_aux k xs)"
ws_class_induct: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and P :: "qtname \<Rightarrow> bool" assumes "class G C = Some c"   and "ws_prog G"   and "\<And>co. class G Object = Some co \<Longrightarrow> P Object"   and "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object; P (super c)\<rbrakk> \<Longrightarrow> P C" shows "P C"
FactRing_def: fixes R :: "('a, 'b) ring_scheme"   and I :: "'a set" shows "R Quot I = \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, one = I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>, zero = I, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
runit_simps_3: fixes f :: "'a" assumes "S.ide f" shows "S.trg (S.runit f) = S.trg f"
simps_1: fixes p :: "assertion" shows "top_ass (Rib p) = Ribbon p"
scale_prod: fixes x :: "'a"   and a :: "'b"   and b :: "'c" shows "local.scale x (a, b) = (x *a a, x *b b)"
sign_compose: fixes p :: "'c \<Rightarrow> 'c"   and q :: "'c \<Rightarrow> 'c" assumes "permutation p"   and "permutation q" shows "sign (p \<circ> q) = sign p * sign q"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (src a)"
run_get_option: fixes get :: "('f \<Rightarrow> 'd) \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> ('e, 'd) optionT" shows "run_option (get_option get f) = get (\<lambda>s. run_option (f s))"
f2: shows "rk {Q, A', R, A} \<le> 3"
in_pair_minus_pairs: fixes p :: "'e \<times> 'e"   and A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" shows "(p \<in>\<^sub>p A -\<^sub>p B) = (p \<in>\<^sub>p A \<and> p \<notin>\<^sub>p B)"
collapse_def: fixes s :: "nat \<Rightarrow> 'b" shows "\<natural> s \<equiv> \<lambda>n. next n s 0"
Diff_disjoint: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> (B - A) = {}"
intro: fixes M :: "'c set set" assumes "\<And>A. range A \<subseteq> M \<Longrightarrow> \<Union> (range A) \<in> M" shows "sigma_algebra_axioms M"
path_concatenation: fixes x :: "'a"   and y :: "'a" assumes "forward_terminating_path x"   and "backward_terminating_path y"   and "end_points x = start_points y"   and "x ; 1 \<cdot> (x\<^sup>T ; 1 + y ; 1) \<cdot> y\<^sup>T ; 1 = 0" shows "path (x + y)"
vsubstAbs_substAbs_compose1: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "good Y" shows "A $[Y / y]_ys $[y1 // y]_ys = A $[Y #[y1 // y]_ys / y]_ys"
Tinv_mpt: assumes "I.invertible_qmpt" shows "mpt M I.Tinv"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. satC \<xi> c"
case_prodE: fixes c :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a \<times> 'b" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'a" and y :: "'b" where "p = (x, y)" and "c x y"
K_2: shows "1 - alpha \<le> PS.prob {x \<in> space M. 1 \<le> (\<Sum>i = 1..<K. h ((T ^^ i) x))}"
has_densityD_1: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "has_density M N f" shows "f \<in> borel_measurable N"
ta_bisim_init_fin_bisim_ThreadFinishAction: shows "ta_bisim init_fin_bisim \<lbrace>ThreadFinishAction\<rbrace> \<lbrace>ThreadFinishAction\<rbrace>"
length_filter_le: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "length (filter P xs) \<le> length xs"
leg1_in_hom_2: shows "\<guillemotleft>r\<^sub>0s\<^sub>1.p\<^sub>1 : r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>"
induct: fixes P :: "('d \<Rightarrow> 'e \<Rightarrow> 'f) \<Rightarrow> 'd list \<Rightarrow> 'e list \<Rightarrow> bool"   and a0 :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a1 :: "'d list"   and a2 :: "'e list" assumes "\<And>f x xs y ys. P f xs ys \<Longrightarrow> P f (x # xs) (y # ys)"   and "\<And>uu_ uw_. P uu_ [] uw_"   and "\<And>uu_ uv_. P uu_ uv_ []" shows "P a0 a1 a2"
\<epsilon>_dom: shows "C.inverse_arrows (\<epsilon> (C.dom f)) (C.inv (\<epsilon> (C.dom f)))"
field_one_plus_frac2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier K"   and "b \<in> carrier K"   and "a \<plusminus> b \<noteq> \<zero>" shows "1\<^sub>r \<plusminus> -\<^sub>a a \<cdot>\<^sub>r (a \<plusminus> b)\<^bsup>\<hyphen> K\<^esup> = b \<cdot>\<^sub>r (a \<plusminus> b)\<^bsup>\<hyphen> K\<^esup>"
diagonalize_2x2_def: fixes A :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" shows "diagonalize_2x2 A = (if A $h 0 $h 0 = (0::'b) then if A $h 0 $h 1 \<noteq> (0::'b) then let A' = interchange_columns A 0 1; Q' = interchange_columns (Finite_Cartesian_Product.mat (1::'b)) 0 1 in diagonalize_2x2_aux (Finite_Cartesian_Product.mat (1::'b), A', Q') else if A $h 1 $h 0 \<noteq> (0::'b) then let A' = interchange_rows A 0 1; P' = interchange_rows (Finite_Cartesian_Product.mat (1::'b)) 0 1 in diagonalize_2x2_aux (P', A', Finite_Cartesian_Product.mat (1::'b)) else (Finite_Cartesian_Product.mat (1::'b), A, Finite_Cartesian_Product.mat (1::'b)) else diagonalize_2x2_aux (Finite_Cartesian_Product.mat (1::'b), A, Finite_Cartesian_Product.mat (1::'b)))"
clock_simRels_def: shows "clock_simRels \<equiv> \<lambda>a. rel_ext (\<lambda>uu_. \<exists>X X' s s'. uu_ = ((X, s), X', s') \<and> X = X' \<and> {s, s'} \<subseteq> X \<and> envObs a s = envObs a s')"
cases: fixes a :: "('b com option list \<times> 'b) list" assumes "a \<in> par_cptn" obtains P :: "'b com option list" and s :: "'b" where "a = [(P__, s__)]"   | P :: "'b com option list" and t :: "'b" and xs :: "('b com option list \<times> 'b) list" and s :: "'b" where "a = (Pa__, sa__) # (Pa__, t__) # xs__" and "(Pa__, t__) # xs__ \<in> par_cptn"   | P :: "'b com option list" and s :: "'b" and Q :: "'b com option list" and t :: "'b" and xs :: "('b com option list \<times> 'b) list" where "a = (P, s) # (Q, t) # xs" and "(P, s) -pc\<rightarrow> (Q, t)" and "(Q, t) # xs \<in> par_cptn"
measurable_restrict: fixes I :: "'b set"   and X :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'c measure"   and M :: "'b \<Rightarrow> 'd measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> N \<rightarrow>\<^sub>M M i" shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
in_arcs_G: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b" assumes "subdiv_step G rev_G H rev_H u v w uv uw vw" shows "{uv, rev_G uv} \<subseteq> arcs G"
env_restr_eqvt: fixes \<pi> :: "perm"   and d :: "'c set"   and m :: "'c \<Rightarrow> 'd" shows "\<pi> \<bullet> m f|` d = (\<pi> \<bullet> m) f|` (\<pi> \<bullet> d)"
1: shows "real_of_rat \<bar>x\<bar> \<le> real_of_rat (rat_of_nat (length (map of_int_hom.vec_hom fs)) * gs.N ^ (3 * Suc (length (map of_int_hom.vec_hom fs))))"
388: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) \<sqinter> - x = - x"
rel_fd1_fd2: shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"
closed_singleton: fixes "open" :: "'c set \<Rightarrow> bool"   and a :: "'c" assumes "class.t1_space open" shows "topological_space.closed open {a}"
coeff_liftPoly: fixes b :: "nat"   and i :: "nat"   and mp :: "'b mpoly"   and x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (liftPoly b i mp) x = (if \<exists>j\<in>{b..<b + i}. 0 < lookup x j then 0::'b else MPoly_Type.coeff mp (higherPowers b i x))"
upower_rep_eq: fixes P :: "'b hrel"   and i :: "nat" shows "\<lbrakk>power.power II (;;) P i\<rbrakk>\<^sub>e = (\<lambda>b. b \<in> {p. \<lbrakk>P\<rbrakk>\<^sub>e p} ^^ i)"
is_extensional: fixes f :: "'a" assumes "\<not> V.arr f" shows "cod f = V.null"
bij_lens_equiv_id: fixes X :: "'d \<Longrightarrow> 'e" shows "bij_lens X = X \<approx>\<^sub>L 1\<^sub>L"
predicate2I: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
finite_Union_subsets: fixes A :: "'b set set"   and b :: "'b set" assumes "\<forall>a\<in>A. a \<subseteq> b"   and "finite b" shows "finite (\<Union> A)"
arrow_of_spans_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" shows "arrow_of_spans_axioms C \<mu> \<equiv> partial_magma.in_hom C (Chn \<mu>) (span_in_category.apex C (Dom \<mu>)) (span_in_category.apex C (Cod \<mu>)) \<and> C (Leg0 (Cod \<mu>)) (Chn \<mu>) = Leg0 (Dom \<mu>) \<and> C (Leg1 (Cod \<mu>)) (Chn \<mu>) = Leg1 (Dom \<mu>)"