hcju/isabelleps huggingface dataset:
length_Suc_conv: fixes xs :: "'c list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
inj_imp_image_ring_iso: fixes f :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme" assumes "inj_on f (carrier R)" shows "f \<in> ring_iso R (image_ring f R)"
supp_perm_eq_test: fixes x :: "'e"   and p :: "perm" assumes "supp x \<sharp>* p" shows "p \<bullet> x = x"
sorted_remove1: fixes cmp :: "'b comparator"   and xs :: "'b list"   and x :: "'b" assumes "Sorting_Algorithms.sorted cmp xs" shows "Sorting_Algorithms.sorted cmp (remove1 x xs)"
must_wait_iff: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and lt :: "'e + 'f + 'f"   and Ts :: "'f set" shows "must_wait s t lt Ts = (case lt of Inl l \<Rightarrow> \<exists>t'\<in>Ts. t \<noteq> t' \<and> has_lock (locks s $ l) t' | Inr (Inl t') \<Rightarrow> not_final_thread s t' \<and> t' \<in> Ts | Inr (Inr t') \<Rightarrow> all_final_except s Ts \<and> t' \<notin> interrupts s)"
gather_by_strictness: shows "(\<Sum>g\<in>local.F_strict. \<Sum>(x, y)\<in>local.Z_good g. local.weight x y) = (\<Sum>g\<in>local.F_strict. Determinant.det (submatrix A UNIV (g ` {0..<n})) * Determinant.det (submatrix B (g ` {0..<n}) UNIV))"
cmp_simps'_1: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (\<Phi>\<^sub>G (\<mu>, \<nu>))"
S4ClerkUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [MClkNext memCh crCh cst p]_(c p) \<and> $S4 rmhist p \<longrightarrow> unchanged c p"
mult_ac_3: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b * (a * c) = a * (b * c)"
not_MOST: fixes P :: "'b \<Rightarrow> bool" shows "(\<not> (\<forall>\<^sub>\<infinity>x. P x)) = (\<exists>\<^sub>\<infinity>x. \<not> P x)"
inner_verts_conv: fixes p :: "('a \<times> 'a) list"   and u :: "'a" shows "inner_verts p = butlast (tl (pawalk_verts u p))"
IH: shows "input_sizes (deep_model'_l (tl rs)) = replicate (2 ^ (length (tl rs) - 1)) (last rs)"
interchange_rows_transpose: fixes A :: "(('b, 'd) vec, 'c) vec"   and a :: "'d"   and b :: "'d" shows "interchange_rows (Finite_Cartesian_Product.transpose A) a b = Finite_Cartesian_Product.transpose (interchange_columns A a b)"
wp_cond_var: fixes p :: "'a"   and x :: "'a"   and y :: "'a"   and q :: "'a" shows "wp (if p then x else y fi) q = (ad p + wp x q) \<cdot> (d p + wp y q)"
af\<^sub>G_decompose_2: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and w :: "'a set list" shows "af\<^sub>G (\<phi> or \<psi>) w = af\<^sub>G \<phi> w or af\<^sub>G \<psi> w"
igSwapBinpIPresIGWlsBinp_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSwapBinpIPresIGWlsBinp MOD \<equiv> \<forall>zs z1 z2 delta binp. igWlsBinp MOD delta binp \<longrightarrow> igWlsBinp MOD delta (igSwapBinp MOD zs z1 z2 binp)"
step_1: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []" shows "exhaustiveUnitPropagate_dom (applyUnitPropagate state')"
min: fixes x :: "'a" assumes "x \<in> vertices G" shows "(x, x) \<in> getRel idt G"
nth_append_length_plus: fixes xs :: "'c list"   and ys :: "'c list"   and n :: "nat" shows "(xs @ ys) ! (length xs + n) = ys ! n"
compl_eq_compl_iff: fixes x :: "'c"   and y :: "'c" shows "(NOT x = NOT y) = (x = y)"
infinity_ileE: fixes m :: "nat"   and R :: "bool" assumes "\<infinity> \<le> enat m" shows "R"
chg_map_upd: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "m a = Some b" shows "chg_map f a m = m(a \<mapsto> f b)"
cod_simps_2: shows "f.cod.leg0 = fgh.\<mu>.leg0"
size_list_overloaded_def: shows "length \<equiv> size_list (\<lambda>_. 0)"
weak_universalD: fixes t :: "'e itself"   and R :: "('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph"   and G\<^sub>1 :: "('f, 'h) labeled_graph"   and G\<^sub>2 :: "('f, 'h) labeled_graph"   and f\<^sub>1 :: "('g \<times> 'h) set"   and f\<^sub>2 :: "('g \<times> 'h) set"   and G :: "('f, 'e) labeled_graph"   and h\<^sub>1 :: "('g \<times> 'e) set"   and h\<^sub>2 :: "('h \<times> 'e) set" assumes "weak_universal t R G\<^sub>1 G\<^sub>2 f\<^sub>1 f\<^sub>2"   and "graph_homomorphism (snd R) G h\<^sub>1"   and "graph_homomorphism G\<^sub>1 G h\<^sub>2"   and "f\<^sub>1 O h\<^sub>2 \<subseteq> h\<^sub>1" shows "\<exists>h. graph_homomorphism G\<^sub>2 G h \<and> h\<^sub>2 \<subseteq> h"
elims: fixes x :: "'b tree"   and y :: "'b tree" assumes "del_lo x = y" obtains "x = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | l :: "'b tree" and a :: "'b" and r :: "'b tree" where "x = \<langle>l, a, r\<rangle>" and "y = merge l r"
degree_aform_err_exp_aform_err: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "exp_aform_err p X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
unit_disc_fix_unit_circle_fix: fixes M :: "moebius" assumes "unit_disc_fix M" shows "unit_circle_fix M"
atan2_def: fixes y :: "real"   and x :: "real" shows "atan2 y x = (if 0 < x then arctan (y / x) else if x < 0 then if 0 < y then arctan (y / x) + pi else arctan (y / x) - pi else if 0 < y then pi / 2 else if y < 0 then - pi / 2 else 0)"
b2_leadsto_g2: shows "\<turnstile> \<box>[(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2) \<and> SF(N2)_(x, y, sem, pc1, pc2) \<and> \<box>#True \<longrightarrow> (pc2 = #b \<leadsto> pc2 = #g)"
range_ext: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<forall>i. \<forall>x\<in>{0..<i}. f x = g x" shows "f = g"
intros_16: fixes t :: "t"   and v :: "v"   and s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and k :: "i32"   and off :: "nat"   and mem' :: "mem"   and vs :: "v list"   and a :: "nat" assumes "types_agree t v"   and "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "store m (nat_of_int k) off (bits v) (t_length t) = Some mem'" shows "\<lparr>s;vs;[$C ConstInt32 k, $C v, $Store t None a off]\<rparr> \<leadsto>_ i \<lparr>s\<lparr>s.mem := (s.mem s)[j := mem']\<rparr>;vs;[]\<rparr>"
halfspace_simps_2: fixes sctn :: "'e sctn" shows "sabove_halfspace sctn = Collect (gt_halfspace sctn)"
pair_less_iff1: fixes x :: "nat"   and y :: "nat"   and z :: "nat" shows "(((x, y), x, z) \<in> pair_less) = (y < z)"
mpoly_Const_power: fixes a :: "'b"   and n :: "nat" shows "Const (a ^ n) = Const a ^ n"
exec_move_TryI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and C :: "String.literal"   and V :: "nat"   and e' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (try e catch(C V) e') h s ta h' s'"
measurable_PiM_single': fixes I :: "'d set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and N :: "'e measure"   and M :: "'d \<Rightarrow> 'f measure" assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> N \<rightarrow>\<^sub>M M i"   and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" shows "(\<lambda>\<omega> i. f i \<omega>) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
simps_12: fixes v :: "cmsg"   and vb :: "cmsg" shows "catomic (cExp v vb) = False"
zero_fract_def: shows "0 \<equiv> abs_fract (0::'a, 1::'a)"
norm_pres_extensions_def: fixes E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" shows "norm_pres_extensions E p F f = {graph H h |H h. linearform H h \<and> Subspace.subspace H E \<and> Subspace.subspace F H \<and> graph F f \<subseteq> graph H h \<and> (\<forall>x\<in>H. h x \<le> p x)}"
triangulize_det: fixes A :: "'a mat"   and d :: "nat"   and r' :: "'a"   and A' :: "'a mat" assumes "mute_fun mf"   and "det_selection_fun sel_fun"   and "A \<in> carrier_mat d d"   and "local.triangulize A = (r', A')" shows "det A * r' = det A'"
carrier_update_1: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set" assumes "ring R" shows "carrier (R\<lparr>carrier := S\<rparr>) = S"
concrete_category_axioms: shows "concrete_category Obj Hom Id Comp"
A: fixes e :: "'c" shows "(case Inl e of Inl e \<Rightarrow> e) = e"
ir_valid_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_valid P c c' Q \<equiv> \<forall>t t'. Q t t' \<longrightarrow> (\<exists>s s'. P s s' \<and> (c, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t')"
quicksort_by_rel_permutes: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sl :: "'b list"   and xs :: "'b list" shows "mset (quicksort_by_rel R sl xs) = mset (xs @ sl)"
assms: shows "a \<in>\<^sub>\<circ> vinsert \<langle>vcard xs, x\<rangle> xs"
assms_7: shows "ms\<^sub>1 = msx @ mx # tl ms\<^sub>2"
lookup_pair_sort_oalist: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw ko' xs" shows "lookup_pair ko (sort_oalist ko xs) = lookup_pair ko' xs"
establish_invarI: fixes I :: "('v, 'es) state_scheme \<Rightarrow> bool" assumes "on_init param \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"   and "\<And>s s' v0. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s = []; v0 \<in> V0; v0 \<notin> dom (discovered s); s' = new_root v0 s\<rbrakk> \<Longrightarrow> on_new_root param v0 s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s \<noteq> []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s\<rbrakk> \<Longrightarrow> on_finish param u s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<in> dom (finished s); s' = cross_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_cross_edge param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<notin> dom (finished s); s' = back_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_back_edge param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<notin> dom (discovered s); s' = discover u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_discover param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))" shows "is_invar I"
lift_gi_def: fixes f :: "'b \<Rightarrow> bool"   and gi :: "bool \<times> 'b" shows "C2.lift_gi f gi = (if fst gi then inv\<^bsub>C2\<^esub> f (snd gi) else f (snd gi))"
op_set_pick_def: fixes s :: "'a set" shows "op_set_pick s \<equiv> SPEC (\<lambda>x. x \<in> s)"
simps: fixes g :: "(location \<times> location) set"   and a :: "location \<Rightarrow> char list set"   and c :: "actor \<Rightarrow> char list set \<times> char list set"   and l :: "location \<Rightarrow> char list \<times> ((actor \<times> actor set) \<times> nat) set" shows "gra (Lgraph g a c l) = g"
pmf_bernoulli_True: fixes p :: "real" assumes "0 \<le> p"   and "p \<le> 1" shows "pmf (bernoulli_pmf p) True = p"
list_update_append: fixes xs :: "'a list"   and ys :: "'a list"   and n :: "nat"   and x :: "'a" shows "(xs @ ys)[n := x] = (if n < length xs then xs[n := x] @ ys else xs @ ys[n - length xs := x])"
wf_trm_subst_rangeD: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<theta> :: "'c \<Rightarrow> ('a, 'c) Term.term"   and x :: "'c" assumes "intruder_model arity Ana"   and "\<forall>t\<in>subst_range \<theta>. intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t" shows "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity (\<theta> x)"
assms_2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "(\<Psi>, P, P) \<in> Rel"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and list :: "'b list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (list = hd list # tl list \<longrightarrow> P (f2 (hd list) (tl list))))"
assms_1: fixes xset :: "('a, 'b, 'c) formula set['a]"   and c :: "'d" assumes "finite (supp xset)"   and "\<And>x c. x \<in> set_bset xset \<Longrightarrow> P c x" shows "P c (Conj xset)"
AbGroup: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "VectorSpace smult V" shows "AbGroup V"
im_set_mono: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and A1 :: "'c set"   and A2 :: "'c set" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A2"   and "A2 \<subseteq> A" shows "f ` A1 \<subseteq> f ` A2"
subset_iff_less_eq_V: fixes B :: "V set"   and A :: "V set" assumes "small B" shows "(A \<subseteq> B) = (set A \<le> set B \<and> small A)"
override_on_apply_in: fixes a :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "a \<in> A" shows "override_on f g A a = g a"
self_append_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = xs \<cdot> ys) = (ys = \<epsilon>)"
rk_abc\<alpha>\<beta>\<gamma>: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points"   and R :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {Q, R} = 2"   and "rk {R, A, a} = 2"   and "rk {R, B, b} = 2"   and "rk {R, C, c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {a, b, c, \<alpha>, \<beta>, \<gamma>} = 3"
cspan_butterfly_ket: shows "cspan {butterfly (ket i) (ket j) |i j. True} = UNIV"
simps_2: fixes xs :: "'b SList.list"   and n :: "nat" shows "SList.take xs (Suc n) = list_case [] (\<lambda>x l. x # SList.take l n) xs"
abs_conv_abscissa_exp: fixes f :: "'a fds" shows "abs_conv_abscissa (fds_exp f) \<le> abs_conv_abscissa f"
d_2: shows "delta \<longlonglongrightarrow> 0"
a_subid: fixes x :: "'a" shows "ad x \<le> (1::'a)"
adding_emptypair_keeps_SdlHPPB: fixes d :: "'d"   and PP :: "nat set"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set" assumes "SdlHPPB d PP R" shows "SdlHPPB d PP (R \<union> {([], [])})"
simps_2: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and x :: "'b"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "split split" shows "split.del split k x (Node ts t) = (case split ts x of (ls, []) \<Rightarrow> split.rebalance_last_tree k ls (split.del split k x t) | (ls, (sub, sep) # rs) \<Rightarrow> if sep \<noteq> x then split.rebalance_middle_tree k ls (split.del split k x sub) sep rs t else if sub = Leaf then Node (ls @ rs) t else let (sub_s, max_s) = split.split_max k sub in split.rebalance_middle_tree k ls sub_s max_s rs t)"
pointwise_def: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'d \<Rightarrow> 'b"   and m' :: "'d \<Rightarrow> 'c" shows "pointwise P m m' = (\<forall>x. P (m x) (m' x))"
ee_trans: fixes as :: "'a list"   and bs :: "'a list"   and cs :: "'a list" assumes "essentially_equal G as bs"   and "essentially_equal G bs cs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G"   and "set cs \<subseteq> carrier G" shows "essentially_equal G as cs"
assms: shows "x \<in> vars\<^sub>l (l \<cdot>\<^sub>l (\<lambda>x. \<epsilon> (y @ x)))"
ordinal_less_def: fixes x :: "ordinal"   and y :: "ordinal" shows "(x < y) = (\<forall>a\<in>Rep_ordinal x. \<forall>b\<in>Rep_ordinal y. a < b)"
simps_2: fixes u0 :: "'a"   and u :: "'a"   and p :: "'a list" shows "prep_wit_blue u0 (REACH u p) = (if u0 = u then CIRC [] (u0 # p) else REACH u (u0 # p))"
assms_2: shows "chain_name \<in> set (map fst \<Gamma>)"
compact_cball: fixes x :: "'c"   and e :: "real" shows "compact (cball x e)"
at_least_at_most_upper: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>" shows "x \<sqsubseteq>\<^bsub>L\<^esub> b"
lfp_unfold: fixes f :: "'g \<Rightarrow> 'g" assumes "mono f" shows "lfp f = f (lfp f)"
simplex_retraction2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b set" assumes "ChamberComplexRetraction X f"   and "x \<in> f \<turnstile> X" shows "f ` x = x"
NUM_E_1: fixes U :: "(nat \<Rightarrow> nat option) set" assumes "U \<in> NUM" shows "U \<subseteq> \<R>"
steps_star_cycle: fixes us :: "'b list list"   and A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "us \<noteq> []"   and "\<And>x. x \<in> set us \<Longrightarrow> accepts A x" shows "accepts (RegExp2NA.plus A) (concat us)"
cond1: fixes ys :: "nat list"   and b :: "nat list" assumes "ys \<in> fst ` set (c2.gen_check b)" shows "bounded_gen_check (C\<^sub>1 b ys) (B\<^sub>1 b)"
is_expansion_map_final: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list"   and G :: "'a ms"   and b :: "real \<Rightarrow> real" assumes "is_lifting f basis basis'"   and "is_expansion G (b # basis)"   and "length basis' = expansion_level TYPE('b)" shows "is_expansion (map_ms f G) (b # basis')"
\<psi>_\<phi>: fixes f :: "'c"   and b :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : b \<rightarrow> a\<guillemotright>" shows "\<psi> (b, a) (\<phi> (b, a) f) = f"
face_of_Int_supporting_hyperplane_le: fixes S :: "'b set"   and a :: "'b"   and b :: "real" assumes "convex S"   and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b" shows "S \<inter> {x. a \<bullet> x = b} face_of S"
add_mono_right: fixes c :: "'d"   and d :: "'d"   and a :: "'d" assumes "c \<le> d" shows "a + c \<le> a + d"
simps: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op" shows "Smith_Impl.Smith_mxn Smith_1x2 Smith_2x2 div_op A = (if dim_row A = 0 \<or> dim_col A = 0 then (1\<^sub>m (dim_row A), A, 1\<^sub>m (dim_col A)) else if dim_row A = 1 then (1\<^sub>m 1, Smith_Impl.Smith_1xn Smith_1x2 A) else if dim_row A = 2 then Smith_Impl.Smith_2xn Smith_1x2 Smith_2x2 div_op A else if dim_col A = 1 then let (P, S) = Smith_Impl.Smith_nx1 Smith_1x2 A in (P, S, 1\<^sub>m 1) else if dim_col A = 2 then Smith_Impl.Smith_nx2 Smith_1x2 Smith_2x2 div_op A else let A1 = mat_of_row (Matrix.row A 0); A2 = mat_of_rows (dim_col A) (map (Matrix.row A) [1..<dim_row A]); (P1, D1, Q1) = Smith_Impl.Smith_mxn Smith_1x2 Smith_2x2 div_op A2; C = A1 * Q1 @\<^sub>r P1 * A2 * Q1; D = mat_of_rows (dim_col A) [Matrix.row C 0, Matrix.row C 1]; E = mat_of_rows (dim_col A) (map (Matrix.row C) [2..<dim_row A]); (P2, F, Q2) = Smith_Impl.Smith_2xn Smith_1x2 Smith_2x2 div_op D; H = P2 * D * Q2 @\<^sub>r E * Q2; (P_H2, H2) = reduce_column div_op H; (H2_UL, H2_UR, H2_DL, H2_DR) = split_block H2 1 1; (P3, S', Q3) = Smith_Impl.Smith_mxn Smith_1x2 Smith_2x2 div_op H2_DR; S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_row A - 1) 1) S'; P1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P1; P2' = four_block_mat P2 (0\<^sub>m 2 (dim_row A - 2)) (0\<^sub>m (dim_row A - 2) 2) (1\<^sub>m (dim_row A - 2)); P3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P3; Q3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q3 in (P3' * P_H2 * P2' * P1', S, Q1 * Q2 * Q3'))"
fls_inverse_const: fixes a :: "'a" shows "inverse (fls_const a) = fls_const (inverse a)"
inv_in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "D.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>D.inv f : b \<Rightarrow>\<^sub>D a\<guillemotright>"
float_eq_refl: fixes a :: "('a, 'b) IEEE.float" shows "a \<doteq> a = (\<not> IEEE.is_nan a)"
word_ao_absorbs_2: fixes x :: "'a word"   and y :: "'a word" shows "x OR y AND x = x"
assms_2: shows "\<forall>f\<in>relations A. f a = 0"
Int_left_commute: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
lift_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and inp :: "'a \<Rightarrow> 'd option" shows "lift g (lift f inp) = lift (g \<circ> f) inp"
st_succ_def: fixes A :: "('d, 'c) nba"   and a :: "'d"   and g :: "'c \<Rightarrow> nat option"   and P :: "'c set" shows "st_succ A a g P \<equiv> {q \<in> if P = {} then dom g else \<Union> (transition A a ` P). even (the (g q))}"
rev_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "rev (map f xs) = map f (rev xs)"
summable_def: fixes f :: "nat \<Rightarrow> 'c" shows "summable f = (\<exists>s. f sums s)"
not_None_eq: fixes x :: "'c option" shows "(x \<noteq> \<bottom>) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
prefix_length_prefix: fixes ps :: "'b list"   and xs :: "'b list"   and qs :: "'b list" assumes "ps \<le>p xs"   and "qs \<le>p xs"   and "\<^bold>|ps\<^bold>| \<le> \<^bold>|qs\<^bold>|" shows "ps \<le>p qs"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat \<Rightarrow>\<^sub>0 nat, 'b) fmap" shows "fmlookup (swapPoly\<^sub>f x xa xb) = (\<lambda>mon. fmlookup xb (swap0 x xa mon))"
ur: shows "\<exists>!x. root_cond (p, l', r') x"
close_closed: fixes C :: "('a, 'b) form set set" shows "subset_closed (close C)"
add: fixes f :: "'k \<Rightarrow> 'l"   and x :: "'k"   and y :: "'k" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
H'_4: shows "H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}})"
lfp_fusion2_aux: fixes g :: "'d \<rightarrow> 'e"   and f :: "'d \<rightarrow> 'd"   and h :: "'e \<rightarrow> 'e"   and i :: "nat" assumes "g oo f = h oo g" shows "g\<cdot>(iterate i\<cdot>f\<cdot>\<bottom>) = iterate i\<cdot>h\<cdot>(g\<cdot>\<bottom>)"
evaln_nonstrict: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and n :: "nat"   and w :: "vals"   and s' :: "state"   and m :: "nat" assumes "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (w, s')"   and "n \<le> m" shows "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>m\<rightarrow> (w, s')"