hcju/isabelleps huggingface dataset:
replication_number_value: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and x :: "'a" assumes "bibd point_set block_collection u_block_size index"   and "x \<in> point_set" shows "block_collection rep x = index * (int (card point_set) - 1) div (u_block_size - 1)"
mono_sup: fixes f :: "'a multiset \<Rightarrow> 'b"   and A :: "'a multiset"   and B :: "'a multiset" assumes "subset_mset.mono f" shows "f A \<squnion> f B \<le> f (A \<union># B)"
op_map_contains_key: fixes k :: "'a"   and m :: "('a, 'b) fmap" shows "(\<in>#) $ k $ (dom_m $ m) \<equiv> op_fmap_contains_key $' k $' m"
image_vimage_subset: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'b set" shows "f ` f -` A \<subseteq> A"
subspace_add: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set"   and x :: "'d"   and y :: "'d" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
derived_alt_group_const: fixes n :: "nat" assumes "5 \<le> n" shows "derived (alt_group n) (carrier (alt_group n)) = carrier (alt_group n)"
assl_def: fixes abc :: "'e \<times> 'f \<times> 'g" shows "assl abc = ((fst abc, fst (snd abc)), snd (snd abc))"
DSourcesA32_L1: shows "DSources level1 sA32 = {sA23}"
linorder_to_class: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "linorder cmp"   and "\<And>x y. cmp x y = EQUAL \<Longrightarrow> x = y" shows "class.linorder (comp2le cmp) (comp2lt cmp)"
dvd_dvd_smult: fixes a :: "'a"   and b :: "'a"   and f :: "'a poly"   and g :: "'a poly" assumes "a dvd b"   and "f dvd g" shows "Polynomial.smult a f dvd Polynomial.smult b g"
word_of_int_bin_cat_eq_iff: fixes b :: "'c word"   and a :: "'b word"   and d :: "'c word"   and c :: "'b word" assumes "LENGTH('b) + LENGTH('c) \<le> LENGTH('d)" shows "(word_of_int (concat_bit LENGTH('c) (uint b) (uint a)) = word_of_int (concat_bit LENGTH('c) (uint d) (uint c))) = (b = d \<and> a = c)"
assms_1: fixes x :: "'a"   and x' :: "'b" assumes "(x, x') \<in> R"   and "I x'" shows "(c x, c' $ x') \<in> bool_rel"
mult_eq_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
class_name_f_def: fixes cld :: "cld" shows "class_name_f cld = (case cld of cld_def dcl cl fds mds \<Rightarrow> dcl)"
start_non_sdom: fixes i :: "nat"   and res :: "nat list"   and k :: "nat" shows "i < n \<and> start ! i = res \<and> k \<notin> set res \<longrightarrow> non_strict_dominate k i"
Beta_altdef: fixes a :: "'a"   and b :: "'a" shows "Beta a b = Gamma a * Gamma b * rGamma (a + b)"
sq_norm_pos: fixes j :: "nat" assumes "j < length (map of_int_hom.vec_hom fs)" shows "0 < \<parallel>gs1.gso j\<parallel>\<^sup>2"
nat_to_string_neq_nil2: fixes n :: "nat" shows "[] \<noteq> nat_to_string n"
set_cases: fixes e :: "'e"   and a :: "'e option" assumes "e \<in> set_option a" obtains "a = Some e"
meet_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
lim_ereal: fixes f :: "'b \<Rightarrow> real"   and x :: "real"   and net :: "'b filter" shows "((\<lambda>n. ereal (f n)) \<longlongrightarrow> ereal x) net = (f \<longlongrightarrow> x) net"
intro: fixes m_ops :: "('a, 'b, 'c, 'd) map_ops_scheme"   and s_ops :: "('e, 'b, 'f) set_ops_scheme"   and t_ops :: "('e, 'g, 'h) set_ops_scheme" assumes "index_loc m_ops s_ops"   and "StdSet t_ops" shows "build_index_loc m_ops s_ops t_ops"
C: assumes "\<tau> \<Turnstile> A" shows "\<tau> \<Turnstile> B"
float_plus_down_float: fixes x :: "real"   and y :: "real"   and p :: "nat" assumes "x \<in> float"   and "y \<in> float" shows "plus_down p x y \<in> float"
dtor1_ctor1: fixes x :: "('d, 'd JF1, 'd JF2) F1" shows "dtor1 (ctor1 x) = x"
IH: fixes s :: "'a wordinterval" shows "length (partIps s ts_) \<le> length ts_ * 2"
assms_5: shows "ptr \<notin> set |h \<turnstile> get_dom_component ptr'|\<^sub>r"
intro: fixes g :: "'b poly"   and h :: "'b poly" assumes "coprime g h"   and "0 < degree g"   and "0 < degree h" shows "fps_hensel_aux g h"
smooth_on_mult: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x * g x)"
variation_cases_3: fixes x :: "real"   and y :: "real" assumes "x < 0"   and "0 < y" shows "variation x y = 1"
LID\<^sub>SI_2: fixes \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and l' :: "'e"   and v :: "('d, 'e, 'f) val"   and l :: "'e" assumes "\<sigma> l' = Some v"   and "l \<in> LID\<^sub>V v" shows "l \<in> LID\<^sub>S \<sigma>"
take_eq_append_take_take_drop: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "m \<le> n" shows "take n xs = take m xs @ take (n - m) (drop m xs)"
is_red_indE: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F p" shows "(\<exists>f\<in>F. f \<noteq> 0 \<and> lt f adds\<^sub>t lt p) \<or> is_red F (tail p)"
generalized_fw_join_def: fixes l1 :: "('a simple_match \<times> 'b) list"   and l2 :: "('a simple_match \<times> 'c) list" shows "generalized_fw_join l1 l2 \<equiv> concat (map (\<lambda>(m1, a). concat (map (\<lambda>(m2, b). map (\<lambda>u. (u, a, b)) (option2list (simple_match_and m1 m2))) l2)) l1)"
distinct_filter: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "distinct xs" shows "distinct (filter P xs)"
lfinite_LConsI: fixes xs :: "'b llist"   and x :: "'b" assumes "lfinite xs" shows "lfinite (LCons x xs)"
res_mult_eq: fixes x :: "int"   and y :: "int" shows "x \<otimes> y = x * y mod int p"
dom_comp: fixes g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" assumes "Maps.seq g f" shows "Maps.dom (g \<odot> f) = Maps.dom f"
a: shows "\<integral>\<^sup>+ x. ennreal (f x) * nn_transfer_operator gp x \<partial>M < \<infinity>"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (save_restore_instr instr s1)) \<and> t2 = snd (fst (save_restore_instr instr s2))"
Block_undefined: fixes S :: "'c set"   and P :: "'c set set"   and a :: "'c" assumes "partition S P"   and "a \<notin> S" shows "partition.Block S P a = undefined"
prems_4: shows "sorted_wrt (\<lambda>x y. y < x) (\<alpha> # \<alpha>s)"
NotInDom_LstSeqP_Eats: fixes k :: "tm"   and s :: "tm"   and y :: "tm"   and z :: "tm" shows "{NotInDom (SUCC k) s, LstSeqP s k y} \<turnstile> LstSeqP (Eats s (HPair (SUCC k) z)) (SUCC k) z"
eventually_at: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and S :: "'a set" shows "eventually P (at a within S) = (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
refl: fixes a :: "'e multiset" shows "a \<subseteq># a"
proj_set_subset: fixes n :: "'e"   and A :: "('e strand_label \<times> 'f) list" shows "set (proj n A) \<subseteq> set A"
symmetric_P: fixes I :: "'a set"   and J :: "'a set" assumes "\<exists>a b. Q I a b"   and "\<exists>a b. Q J a b"   and "P I J" shows "P J I"
prv_lcnj_imp_inner: fixes \<phi> :: "'fmla"   and \<phi>1s :: "'fmla list"   and \<phi>2s :: "'fmla list" assumes "\<phi> \<in> fmla"   and "set \<phi>1s \<subseteq> fmla"   and "set \<phi>2s \<subseteq> fmla" shows "prv (imp (cnj \<phi> (lcnj (\<phi>1s @ \<phi>2s))) (lcnj (\<phi>1s @ \<phi> # \<phi>2s)))"
mult_ge1_mono_2: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "(1::'b) \<le> b" shows "a \<le> b * a"
map\<^sub>0_def: fixes a :: "'a" shows "SPN.map\<^sub>0 a \<equiv> Span.src (SPN a)"
proj5_invalidate: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat" shows "\<pi>\<^sub>5 (the (invalidate rt dests dip)) = \<pi>\<^sub>5 (the (rt dip))"
div_mult_mult1_if: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "divide (times c a) (times c b) = (if c = zero then zero else divide a b)"
slim_paths_in_slimG: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "(u, p, v) \<in> S.slim_paths" shows "pre_digraph.gen_iapath (with_proj S.slim) (verts3 (with_proj slim)) u p v \<and> p \<noteq> []"
s\<^sub>2: shows "s\<^sub>2 \<in> A"
IH: assumes "boundary_chain (set l)"   and "valid_chain_list l"   and "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>"   and "\<forall>a\<in>set l. case a of (k, a) \<Rightarrow> line_integral_exists F basis a" shows "line_integral_exists F basis (rec_join l)"
G2_nodes: shows "G2_nodes = {n. \<exists>ps. valid_graph.is_path (del_unEdge v w v' G) n ps v'}"
pair_measureI: fixes x :: "'a set"   and A :: "'a measure"   and y :: "'b set"   and B :: "'b measure" assumes "x \<in> sets A"   and "y \<in> sets B" shows "x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
antisymp_on_LEX: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "irreflp_on P A"   and "antisymp_on P A" shows "antisymp_on (LEX P) (SEQ A)"
cases: fixes a :: "'b llist" assumes "lfinite a" obtains "a = LNil"   | xs :: "'b llist" and x :: "'b" where "a = x ## xs" and "lfinite xs"
map_disj_restrict_dom: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" shows "h\<^sub>0 \<bottom> h\<^sub>1 |` (dom h\<^sub>1 - dom h\<^sub>0)"
IH: shows "liminf (\<lambda>n. u (n + k)) = liminf u"
avl_delete_auto_1: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t" shows "avl (delete x t)"
mod_enat_def: fixes a :: "enat"   and b :: "enat" shows "a mod b \<equiv> case a of enat x \<Rightarrow> case b of enat y \<Rightarrow> enat (x mod y) | \<infinity> \<Rightarrow> a | \<infinity> \<Rightarrow> \<infinity>"
semiattractive_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "semiattractive B ((\<sqsubseteq>) \<restriction> A)"
lrbe_def: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec"   and \<rho> :: "'b V_rec \<Rightarrow> 'b set"   and ur :: "('a \<times> 'a) set" assumes "Unwinding SES \<V>" shows "Unwinding.lrbe SES \<V> \<rho> ur \<equiv> \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. reachable SES s \<and> Unwinding.En SES \<V> \<rho> s c \<longrightarrow> (\<exists>s'\<in>S\<^bsub>SES\<^esub>. s c\<longrightarrow>\<^bsub>SES\<^esub> s' \<and> (s, s') \<in> ur)"
simps_1: fixes x :: "complex"   and i :: "nat"   and y :: "complex"   and j :: "nat"   and yjs :: "(complex \<times> nat) list" shows "delete_cnj x i ((y, j) # yjs) = (if x = y then if Suc j = i then yjs else if i < Suc j then (y, j - i) # yjs else delete_cnj x (i - Suc j) yjs else (y, j) # delete_cnj x i yjs)"
awalk_verts_ne_eq: fixes p :: "('a \<times> 'a) list"   and u :: "'a"   and v :: "'a" assumes "p \<noteq> []" shows "pawalk_verts u p = pawalk_verts v p"
times_scalar_id: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and v :: "'b list" assumes "Matrix_Tensor.mult id f" shows "mult.times f id v = v"
project_ensures_D_lemma: fixes G :: "'c program"   and C :: "'c set"   and A :: "'a set"   and B :: "'a set"   and F :: "'a program" assumes "G \<in> stable (C \<inter> extend_set h A - extend_set h B)"   and "F \<squnion> project h C G \<in> project_set h C \<inter> A ensures B"   and "extend h F \<squnion> G \<in> stable C" shows "extend h F \<squnion> G \<in> C \<inter> extend_set h A ensures extend_set h B"
assms_2: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
mem_correct_tracing: shows "mem_correct lookup_trace update_trace (P \<circ> snd)"
instInp_fmla: fixes \<phi> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "t \<in> trm" shows "instInp \<phi> t \<in> fmla"
assms_1: fixes x' :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "(x', e) \<in> set \<Gamma>" shows "ccExp_syn a e[x::=y] G|` S = ccExp_syn a e G|` S"
obs: shows "cball t (min e1 e2) \<subseteq> T"
pochhammer_Suc: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc n) = times (comm_semiring_1.pochhammer times one plus zero a n) (plus a (semiring_1.of_nat one plus zero n))"
neutralSellerVariant: fixes A :: "(integer \<times> 'c) set set"   and b :: "integer \<times> 'c \<Rightarrow> 'd" assumes "\<forall>a\<in>A. finite a \<and> (\<forall>X. X \<in> Range a \<longrightarrow> b (seller, X) = (0::'d))" shows "{sum b a |a. a \<in> A} = {sum b (Outside' {seller} a) |a. a \<in> A}"
fds_abs_converges_subseries: fixes f :: "'a fds"   and s :: "'a"   and P :: "nat \<Rightarrow> bool" assumes "fds_abs_converges f s" shows "fds_abs_converges (fds_subseries P f) s"
f: fixes \<Omega> :: "'a set" assumes "\<Omega> \<in> C" shows "f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N"
simps_14: fixes f11 :: "'a"   and f12 :: "X \<Rightarrow> 'a \<Rightarrow> 'a"   and f13 :: "Y \<Rightarrow> 'b \<Rightarrow> 'a"   and f21 :: "X \<Rightarrow> 'a \<Rightarrow> 'b"   and f22 :: "Y \<Rightarrow> 'b \<Rightarrow> 'b"   and f23 :: "'b"   and x13 :: "Y" shows "rec_X f11 f12 f13 f21 f22 f23 (X.C x13) = f13 x13 (rec_Y f11 f12 f13 f21 f22 f23 x13)"
simps_18: shows "get_axiom ADIGr = DIGraxiom"
map_def: fixes f :: "'c local.term set" shows "\<nu>.map f = (if arr f then mkarr (DOM (cod f)\<^bold>\<down>) \<cdot> local.map f else null)"
fbox_export1: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "ad p +  |x] q =  |d p \<cdot> x] q"
i: shows "\<And>s. P \<Longrightarrow> Q s"
valid_lists_subset: fixes xs :: "Vars.var list"   and S :: "Vars.var set"   and G :: "CoCalls" assumes "xs \<in> valid_lists S G" shows "set xs \<subseteq> S"
pm_of_poly_of_pm: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and x :: "'c" assumes "p \<in> P[{x}]" shows "pm_of_poly x (poly_of_pm x p) = p"
gga_to_list_def: fixes g :: "'G" shows "gga_to_list g \<equiv> (nodes_it g (\<lambda>_. True) (#) [], edges_it g (\<lambda>_. True) (#) [])"
apx_in: fixes Z :: "('a \<Rightarrow> real) set" assumes "Z \<subseteq> V" shows "Approx\<^sub>\<beta> Z \<in> {S. \<exists>U M. S = \<Union> U \<and> U \<subseteq> \<R> \<and> Z \<subseteq> S \<and> (S = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n) \<and> normalized M}"
span_superset: fixes S :: "'d set" shows "S \<subseteq> span S"
rbt_comp_insert: shows "local.rbt_comp_insert = ord.rbt_insert (lt_of_comp c)"
aux_qFresh_ignoreFirst2_def: fixes K :: "'c \<times> 'd \<times> ('a, 'b, 'c, 'd, 'e) qTerm + 'c \<times> 'd \<times> ('a, 'b, 'c, 'd, 'e) qAbs" shows "aux_qFresh_ignoreFirst2 K = (case K of Inl (zs, x, xa) \<Rightarrow> termIn xa | Inr (zs, x, xa) \<Rightarrow> absIn xa)"
monad_envT: assumes "monad return bind" shows "monad local.return_env local.bind_env"
interval_upperbound: fixes a :: "'a"   and b :: "'a" assumes "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" shows "interval_upperbound (cbox a b) = b"
Par2B: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P \<parallel> Q'"
Ds_mono: fixes A :: "'f set option"   and A' :: "'f set option"   and es :: "('f, 'g, 'h) exp list" assumes "A \<sqsubseteq> A'"   and "\<D>s es A" shows "\<D>s es A'"
hyps_3: shows "subdivision_step I rev_I (with_proj H) (swap_in (parcs H)) (u, v, w) (uv, uw, vw)"
eval_cexpr_def: fixes e :: "cexpr"   and \<sigma> :: "nat \<Rightarrow> val"   and v :: "val" shows "eval_cexpr e \<sigma> v = extract_real (cexpr_sem (case_nat v \<sigma>) e)"
norm_triangle_ineq3: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a - norm b\<bar> \<le> norm (minus a b)"
inf_a_def: fixes a :: "'a"   and b :: "'a" shows "inf_a a b = -l (a r\<rightarrow> -r (a l\<rightarrow> b))"
prems: fixes Bs :: "'a tensor list"   and a :: "'a" assumes "\<And>B. B \<in> set Bs \<Longrightarrow> order B = 1"   and "a \<cdot> prod_list Bs = A \<otimes> B" shows "thesis"
assms_3: shows "R \<mu> = (f \<star> \<iota>) \<cdot>\<^sub>S \<a>[f, a, a]"
abs_eq: fixes x :: "(11, 52) IEEE.float" shows "Code_Double.is_nan (abs_double x) = IEEE.is_nan x"
qtable_assign: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and y :: "nat"   and A' :: "nat set"   and Q' :: "'a option list \<Rightarrow> bool"   and f :: "'a option list \<Rightarrow> 'a" assumes "qtable n A P Q X"   and "y < n"   and "insert y A = A'"   and "y \<notin> A"   and "\<And>x'. \<lbrakk>wf_tuple n A' x'; P x'\<rbrakk> \<Longrightarrow> P (restrict A x')"   and "\<And>x. \<lbrakk>wf_tuple n A x; P x; Q x\<rbrakk> \<Longrightarrow> Q' (x[y := Some (f x)])"   and "\<And>x'. \<lbrakk>wf_tuple n A' x'; P x'; Q' x'\<rbrakk> \<Longrightarrow> Q (restrict A x') \<and> x' ! y = Some (f (restrict A x'))" shows "qtable n A' P Q' ((\<lambda>x. x[y := Some (f x)]) ` X)"
finite_sorted_distinct_unique: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less"   and "finite A" shows "\<exists>!xs. set xs = A \<and> linorder.sorted less_eq xs \<and> distinct xs"
quasi_order_def: fixes qle :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "quasi_order qle \<equiv> (\<forall>x. qle x x) \<and> (\<forall>x y z. qle x y \<longrightarrow> qle y z \<longrightarrow> qle x z)"
frameImpIntComposition: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" assumes "\<Psi> \<simeq> \<Psi>'" shows "\<langle>A\<^sub>F, \<Psi> \<otimes> \<Psi>\<^sub>F\<rangle> \<hookrightarrow>\<^sub>F \<langle>A\<^sub>F, \<Psi>' \<otimes> \<Psi>\<^sub>F\<rangle>"
vrestr_comp: fixes g :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "g \<in> M \<rightarrow>\<^sub>M M" shows "(f \<circ> g) --` A = g --` f --` A"
FinalAllows_subseteq_in_doubt_allow: fixes \<gamma> :: "'i common_primitive \<Rightarrow> 'a \<Rightarrow> bool"   and \<beta> :: "'i common_primitive \<Rightarrow> 'a \<Rightarrow> ternaryvalue"   and \<Gamma> :: "char list \<Rightarrow> 'i common_primitive rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs" shows "_Collect p (\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow) \<subseteq> _Collect p ((\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow)"
supp_at_base: fixes a :: "'d" shows "supp a = {atom a}"
save_restore_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (save_restore_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
object_ptr_kinds_M_eq2_h2: shows "|h2 \<turnstile> object_ptr_kinds_M|\<^sub>r = |h3 \<turnstile> object_ptr_kinds_M|\<^sub>r"
vdiff_le: fixes v :: "nat list"   and w :: "nat list"   and x :: "nat list" assumes "v \<le>\<^sub>v w"   and "length v = length x" shows "v -\<^sub>v x \<le>\<^sub>v w"
Int_Un_eq_3: fixes S :: "'f set"   and T :: "'f set" shows "S \<union> S \<inter> T = S"
cycle_edges_edges: fixes G :: "('a, 'b) graph_scheme"   and p :: "'a list" assumes "cycle G p" shows "set (cycle_edges p) \<subseteq> \<^bold>E\<^bsub>G\<^esub>"
decr_grading_plus: fixes d :: "'c \<Rightarrow> nat"   and n :: "nat"   and s :: "'c"   and t :: "'c" assumes "hom_grading d" shows "decr_grading d n (s + t) = decr_grading d n s + decr_grading d n t"
distinct_5: fixes x21 :: "'a"   and x22 :: "('a, 'b) Expression"   and x23 :: "'a"   and x31 :: "'a"   and x32 :: "('a, 'b) Expression"   and x33 :: "('a, 'b) Expression"   and x34 :: "'a" shows "Vx : x21 \<turnstile> x22 : x23 \<noteq> Vx : x31 \<turnstile> x32\<equiv>x33 : x34"
match_simps_1: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "term"   and u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" shows "match (t\<^sub>1 $ t\<^sub>2) (app u\<^sub>1 u\<^sub>2) = match t\<^sub>1 u\<^sub>1 \<bind> (\<lambda>env\<^sub>1. match t\<^sub>2 u\<^sub>2 \<bind> (\<lambda>env\<^sub>2. Some (env\<^sub>1 ++\<^sub>f env\<^sub>2)))"
coeff_mult: fixes p :: "'b poly"   and q :: "'b poly"   and n :: "nat" shows "poly.coeff (p * q) n = (\<Sum>i\<le>n. poly.coeff p i * poly.coeff q (n - i))"
commute: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "linordered_monoid.max a b = linordered_monoid.max b a"
assms_5: shows "\<bar>c - c'\<bar> \<le> \<delta>_c"
euler_witnessI2: fixes a :: "int"   and b :: "int"   and k :: "nat"   and n :: "nat"   and x :: "int" assumes "[a \<noteq> b] (mod int k)"   and "k dvd n"   and "euler_liar x n \<Longrightarrow> [Jacobi x (int n) = a] (mod int k)"   and "euler_liar x n \<Longrightarrow> [x ^ ((n - 1) div 2) = b] (mod int k)" shows "euler_witness x n"
bin_nth_sc_gen: fixes n :: "nat"   and b :: "bool"   and w :: "int"   and m :: "nat" shows "bit (bin_sc n b w) m = (if m = n then b else bit w m)"
is_constant_seq_def: fixes R :: "('c, 'd) partial_object_scheme"   and s :: "nat \<Rightarrow> 'c" shows "is_constant_seq R s = (\<exists>x\<in>carrier R. s = constant_function UNIV x)"
ex_fin_conv: fixes A :: "'b fset" shows "(\<exists>x. x |\<in>| A) = (A \<noteq> {||})"
simps_1: fixes c :: "'d" shows "BVO (OVar c) = UNIV"
path2_split_first_prop: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and P :: "'node \<Rightarrow> bool" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "\<exists>x\<in>set ns. P x" obtains m' :: "'node" and ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>m'" and "P m'" and "\<forall>x\<in>set (butlast ns'). \<not> P x" and "prefix ns' ns"
interchange: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and \<tau> :: "'a"   and \<sigma> :: "'a" assumes "S.seq \<nu> \<mu>"   and "S.seq \<tau> \<sigma>"   and "\<mu> \<star> \<sigma> \<noteq> null" shows "\<nu> \<cdot>\<^sub>S \<mu> \<star> \<tau> \<cdot>\<^sub>S \<sigma> = (\<nu> \<star> \<tau>) \<cdot>\<^sub>S (\<mu> \<star> \<sigma>)"
initial_empty_rels: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> ('e \<Rightarrow> 'f option)) list"   and m :: "'g"   and \<S> :: "nat \<Rightarrow> bool option" assumes "initial (ts, m, \<S>)" shows "\<forall>\<R>\<in>set (map released ts). \<R> = Map.empty"
deadlocked: fixes t :: "'t" assumes "not_final_thread s t" shows "t \<in> deadlocked s"