hcju/isabelleps huggingface dataset:
distinct_2: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "\<langle>x21, x22, x23\<rangle> \<noteq> \<langle>\<rangle>"
equiv'_def: shows "Pre.equiv' \<equiv> \<lambda>a b. a \<sim> b \<and> PA a \<and> PB b"
phils_state_update_def: fixes ps :: "nat list"   and e :: "dining_event" shows "Up\<^sub>P ps e = (let i = phil e in if i = 0 then ps[i := Up\<^sub>l\<^sub>p (ps ! i) e] else ps[i := Up\<^sub>r\<^sub>p i (ps ! i) e])"
hensel_main: fixes C :: "int poly"   and Fs :: "int poly factor_tree"   and Gs :: "int poly list" assumes "C =m \<Prod>\<^sub># (factors_of_factor_tree Fs)"   and "\<And>F. F \<in># factors_of_factor_tree Fs \<Longrightarrow> Mp F = F \<and> monic F"   and "hensel_main C Fs = Gs"   and "monic C"   and "pn.Mp C = C"   and "square_free_m C"   and "\<And>f t. \<lbrakk>t \<in> sub_trees Fs; factor_node_info t = f\<rbrakk> \<Longrightarrow> f = Mp (\<Prod>\<^sub># (factors_of_factor_tree t))" shows "pn.eq_m C (prod_list Gs) \<and> factors_of_factor_tree Fs = mset (map Mp Gs) \<and> (\<forall>G. G \<in> set Gs \<longrightarrow> monic G \<and> pn.Mp G = G)"
DERIV_image_chain: fixes f :: "'a \<Rightarrow> 'a"   and Da :: "'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and s :: "'a set"   and Db :: "'a" assumes "(f has_field_derivative Da) (at (g x) within g ` s)"   and "(g has_field_derivative Db) (at x within s)" shows "(f \<circ> g has_field_derivative Da * Db) (at x within s)"
pmaxsimpchain_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "pgallery (x # xs)" shows "pgallery xs"
IH: shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0_ \<I>_ (decomp_rm\<^sub>e\<^sub>s\<^sub>t S_)"
enum_set_get_one_Some: fixes S :: "'b set"   and ss :: "'b list"   and x :: "'b" assumes "S \<subseteq> set ss"   and "enum_set_get_one ss S = Some x" shows "x \<in> S"
Cd_Above: fixes d :: "'d"   and X :: "'x set" shows "Cd d X = Above (Pd d) (X \<inter> Field (Pd d)) \<inter> X"
pcompose_eq_0: fixes q :: "'a poly"   and p :: "'a poly" assumes "degree q \<noteq> 0" shows "(p \<circ>\<^sub>p q = 0) = (p = 0)"
impl_adjoin: fixes P :: "'d upred"   and Q :: "'d upred"   and R :: "'d upred" shows "((P \<Rightarrow> Q) \<and> R) = ((P \<and> R \<Rightarrow> Q \<and> R) \<and> R)"
strict_order_iff: fixes a :: "'a word"   and b :: "'a word" shows "(a <s b) = (a = signed.min a b \<and> a \<noteq> b)"
G_empty_def: shows "G_empty \<equiv> wgraph_by_map.empty RBT_Set.empty"
monomial_0_iff: fixes t :: "'d"   and c :: "'e" shows "(monomial c t = 0) = (c = (0::'e))"
xfer_run_aux: fixes cs :: "nat \<Rightarrow> 'c" assumes "C.is_run cs" shows "\<exists>as. (\<forall>i. R (as i) (cs i)) \<and> A.is_run as"
normal_def: fixes sctn :: "'c sctn" shows "normal sctn = (case sctn of Sctn x1 x2 \<Rightarrow> x1)"
drop_lem: fixes n :: "nat"   and l :: "'c list"   and g :: "'c list" assumes "drop n l = drop n g"   and "length l = length g"   and "n < length g" shows "l ! n = g ! n"
dominated_convergence_integrable: fixes S :: "'c set"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>k. f k absolutely_integrable_on S"   and "h integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (g x) \<le> h x"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x" shows "g integrable_on S"
reflexive: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<leadsto>\<guillemotleft>Rel\<guillemotright> P"
DiaN: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and p :: "('c, 'b) fm"   and j :: "'b" assumes "Hintikka A H"   and "(\<^bold>\<not> (\<^bold>\<diamond> p)) at i in' H"   and "(\<^bold>\<diamond> Nom j) at i in' H" shows "(\<^bold>\<not> (\<^bold>@ j p)) at i in' H"
add_cancel_right_right: fixes a :: "'b"   and b :: "'b" shows "(a = a + b) = (b = (0::'b))"
int_distrib_1: fixes z1 :: "int"   and z2 :: "int"   and w :: "int" shows "(z1 + z2) * w = z1 * w + z2 * w"
neutral: fixes A :: "'a multiset" assumes "\<forall>x\<in>#A. x = \<^bold>1" shows "F A = \<^bold>1"
streamValue2: fixes D :: "nat"   and z :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and x :: "'a" assumes "\<forall>j\<le>D + z. str (t + j) = x" shows "\<forall>j\<le>D. str (t + j + z) = x"
order_on_defs_4: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
local_help': fixes a :: "'c list"   and r :: "'c"   and b :: "'c list"   and c :: "'c list"   and d :: "'c list" assumes "a @ r # b = c @ r # d"   and "r \<notin> set a"   and "r \<notin> set b" shows "a = c \<and> b = d"
simps_3: fixes vref :: "ref"   and l :: "dag"   and vref1 :: "ref"   and r :: "dag" shows "bdt (Node Tip vref (Node l vref1 r)) = Map.empty"
LTL_FG_suffix: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "suffix i w \<Turnstile> F G \<phi> = w \<Turnstile> F G \<phi>"
sup_bot_left: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b" assumes "class.bounded_semilattice_sup_bot (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (bot::'b)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (bot::'b) (x::'b) = x"
lexordp_conv_lexordp_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "ord.lexordp less xs ys = (ord.lexordp_eq less xs ys \<and> \<not> ord.lexordp_eq less ys xs)"
Cod_cod: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f" shows "J_C.Cod (J_C.cod f) = J_C.Cod f"
tests_input_equality: fixes xb :: "nat"   and G :: "vname gexp list" shows "(\<exists>v. Eq (V (I xb)) (L v) \<in> set G) = (1 \<le> length (filter (tests_input_equality xb) G))"
span_scale: fixes x :: "'b"   and S :: "'b set"   and c :: "complex" assumes "x \<in> cspan S" shows "c *\<^sub>C x \<in> cspan S"
sporadic_on_e1: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>"
\<R>_div_def: fixes \<omega> :: "('c \<Rightarrow> 'a) set stream" shows "\<R>_div \<omega> \<equiv> \<forall>x\<in>\<X>. (\<forall>i. (\<exists>j\<ge>i. zero x (\<omega> !! j)) \<and> (\<exists>j\<ge>i. \<not> zero x (\<omega> !! j))) \<or> (\<exists>i. \<forall>j\<ge>i. unbounded x (\<omega> !! j))"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_monoid_mult_hom hom"   and "bijective hom" shows "monoid_mult_isom hom"
top_left_normalize: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "bounded_quasi_semilattice f top bot normalize" shows "f top a = normalize a"
split_sel: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (option = Some (the option) \<longrightarrow> P (f2 (the option))))"
GF_advice_monotone: fixes X :: "'b ltln set"   and Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "X \<subseteq> Y"   and "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>" shows "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<nu>"
return_id: fixes f :: "('a, 'b, 'c) prog" shows "Heap_Error_Monad.bind f return = f"
has_fields_Object: fixes P :: "'a prog"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "FDTs = map (\<lambda>(F, b, T). ((F, Object), b, T)) fs" shows "P \<turnstile> Object has_fields FDTs"
comp_assoc_assoc'_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "V (bicategory.\<a>' V H \<a> src trg f g h) (\<a> f g h) = H (H f g) h"
invariant: fixes a :: "'a"   and b :: "'a" assumes "PA a"   and "A a b" shows "PA b"
eventually_cofinite: fixes P :: "'a \<Rightarrow> bool" shows "Alm_all P = finite {x. \<not> P x}"
irreducible\<^sub>d_dvd_smult: fixes p :: "'b poly"   and q :: "'b poly" assumes "0 < degree p"   and "irreducible\<^sub>d q"   and "p dvd q" shows "\<exists>c. c \<noteq> (0::'b) \<and> q = Polynomial.smult c p"
mC_C_partCompat_cont: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "mC_C theta c d s t P F"   and "sym theta"   and "trans theta" shows "partCompat P theta (local.cont c s)"
nAct_finite: fixes n :: "enat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "cnf llist" assumes "n \<noteq> \<infinity>" shows "\<exists>n'. dynamic_component.nAct tCMP active c n t = enat n'"
rev_simp_invert: fixes xs :: "'b list"   and x :: "'b"   and zs :: "'b list" shows "(xs @ [x] = rev zs) = (zs = x # rev xs)"
eval_fm_rename: fixes k' :: "name"   and A :: "fm"   and e :: "(name, hf) finfun"   and k :: "name"   and x :: "hf" assumes "atom k' \<sharp> A" shows "eval_fm (finfun_update e k x) A = eval_fm (finfun_update e k' x) ((k' \<leftrightarrow> k) \<bullet> A)"
A: shows "jordan_block n (0::'a) ^\<^sub>m k \<in> carrier_mat n n"
Seed_holds_edges_disj: fixes p :: "nat" shows "edges_disj Seed\<^bsub>p\<^esub>"
sum_upto_cong: fixes f :: "nat \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'b"   and n :: "real"   and n' :: "real" assumes "\<And>n. 0 < n \<Longrightarrow> f n = f' n"   and "n = n'" shows "sum_upto f n = sum_upto f' n'"
Loop_def: fixes nil :: "'a"   and "next" :: "('a \<times> 'a) set"   and root :: "'a" assumes "graph nil next" shows "graph.Loop nil root next \<equiv> {(X, mrk). finite (- mrk) \<and> finite X \<and> X \<subseteq> mrk \<and> mrk \<subseteq> graph.reach nil next root \<and> graph.reach nil next root \<inter> - mrk \<subseteq> graph.path next X mrk}"
en_eq_3: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<^bold>\<diamond>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>x,F\<rbrace> in v]"
set_takeWhileD: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "x \<in> ran\<^sub>l (takeWhile P xs)" shows "x \<in> ran\<^sub>l xs \<and> P x"
apx_subset: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and Z :: "('d \<Rightarrow> real) set"   and k :: "'d \<Rightarrow> nat" assumes "Beta_Regions' X v n not_in_X" shows "Z \<subseteq> Beta_Regions'.Approx\<^sub>\<beta> X k v n not_in_X Z"
divide_le_cancel: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a c) (divide b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
lunit_naturality: fixes f :: "'a" assumes "arr f" shows "C.lunit (cod f) \<cdot> C.tensor C.unity f = f \<cdot> C.lunit (local.dom f)"
min_elems_ex: fixes t :: "'b list"   and A :: "'b list set" assumes "t \<in> A" shows "\<exists>t' r. t = t' @ r \<and> t' \<in> min_elems A"
closed_irreducible_separator: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "closed S"   and "\<not> connected_component (- S) a b" obtains T :: "'a set" where "T \<subseteq> S" and "closed T" and "T \<noteq> {}" and "\<not> connected_component (- T) a b" and "\<And>U. U \<subset> T \<Longrightarrow> connected_component (- U) a b"
QR_decomposition: fixes A :: "((real, 'a) vec, 'b) vec" assumes "rank A = ncols A" shows "A = fst (QR_decomposition A) ** snd (QR_decomposition A) \<and> pairwise orthogonal (columns (fst (QR_decomposition A))) \<and> (\<forall>i. norm (column i (fst (QR_decomposition A))) = 1) \<and> Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A) = mat 1 \<and> vec.independent (columns (fst (QR_decomposition A))) \<and> col_space A = col_space (fst (QR_decomposition A)) \<and> card (columns A) = card (columns (fst (QR_decomposition A))) \<and> invertible (snd (QR_decomposition A)) \<and> upper_triangular (snd (QR_decomposition A))"
\<phi>_uniqueness_2: shows "local.iso f\<^sub>0g\<^sub>1.\<phi>"
poly_deg_monomial: fixes t :: "'b \<Rightarrow>\<^sub>0 'a"   and c :: "'c" shows "poly_deg (monomial c t) = (if c = (0::'c) then 0::'a else deg_pm t)"
gamma1_as_euclid_space_fun: shows "gamma1 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R i + (0, g2 (a + (b - a) * x)))"
atLeastatMost_empty_iff2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "({} = subset_mset.atLeastAtMost a b) = (\<not> a \<subseteq># b)"
id1: shows "mset (map ((!) (bef @ s # aft)) (\<sigma> (f, Suc (length bef + length aft)))) = mset (map ((!) (bef @ s # aft)) nlb) + mset (map ((!) (bef @ s # aft)) lb)"
ivl_disj_un_8: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{..<l} \<union> {l..<u} = {..<u}"
ee1: shows "essentially_equal G (ah # as) (ah # take i as' @ drop (Suc i) as')"
lookup_dehomogenize: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and x :: "'b" assumes "homogeneous p"   and "t \<in> keys p" shows "lookup (dehomogenize x p) (except t {x}) = lookup p t"
proj_lift_poly_syz: fixes j :: "nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and i :: "nat" assumes "j < n" shows "proj_poly j (lift_poly_syz n p i) = (1 when j = i)"
echelon_form_upt_k_bezout_matrix: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and i :: "'f"   and n :: "'f" assumes "echelon_form_upt_k A k"   and "is_bezout_ext bezout"   and "A $ i $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "is_zero_row_upt_k i k A"   and "i < n"   and "k < ncols A" shows "echelon_form_upt_k (bezout_matrix A i n (mod_type_class.from_nat k) bezout ** A) k"
beta_reduces_pair_left: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A \<rightarrow>\<beta>\<^sup>* A'" shows "SimplyTyped.Pair A B \<rightarrow>\<beta>\<^sup>* SimplyTyped.Pair A' B"
relAPP_def: fixes f :: "('c \<times> 'd) set \<Rightarrow> 'b"   and x :: "('c \<times> 'd) set" shows "\<langle>x\<rangle>f \<equiv> f x"
xor_assoc: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "(x \<oplus>\<^bsub>L\<^esub> y) \<oplus>\<^bsub>L\<^esub> z = x \<oplus>\<^bsub>L\<^esub> y \<oplus>\<^bsub>L\<^esub> z"
val_bind_tm: fixes m :: "'c tm"   and f :: "'c \<Rightarrow> 'b tm" shows "Time_Monad.val (m \<bind> f) = (let x = Time_Monad.val m in Time_Monad.val (f x))"
tail_face_cycle_succ: fixes a :: "'b \<times> 'b" assumes "a \<in> arcs (with_proj (list_digraph G_list))" shows "tail (with_proj (list_digraph G_list)) (face_cycle_succ a) = head (with_proj (list_digraph G_list)) a"
iter_isolate: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice) \<sqinter> c\<^sup>\<infinity> = c\<^sup>\<omega> ; d"
simps_1: fixes ao :: "'a option"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "proper_interval_Compl_set_aux ao (x # xs) (y # ys) = (if x < y then \<not> proper_interval ao (Some x) \<and> proper_interval_Compl_set_aux (Some x) xs (y # ys) else if y < x then \<not> proper_interval ao (Some y) \<and> proper_interval_Compl_set_aux (Some y) (x # xs) ys else \<not> proper_interval ao (Some x) \<and> (ys = [] \<longrightarrow> xs \<noteq> []))"
snd_vwb_lens: shows "vwb_lens snd\<^sub>L"
nat_pow_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "x [^] n \<in> carrier G"
embed_cong: fixes r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and r' :: "('d \<times> 'd) set" assumes "\<And>a. a \<in> Field r \<Longrightarrow> f a = g a" shows "embed r r' f = embed r r' g"
rotate15: fixes A15 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and A9 :: "fm"   and A10 :: "fm"   and A11 :: "fm"   and A12 :: "fm"   and A13 :: "fm"   and A14 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A15 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 H)))))))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 (insert A15 H)))))))))))))) \<turnstile> B"
freeSubst: shows "dfree (Tsubst \<theta> \<sigma>)"
LAssRed: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow> \<langle>V:=e',s'\<rangle>"
5: shows "(a, b) \<in> F (v (\<Squnion>x. Y x))"
sappend_def: shows "sappend \<equiv> \<mu> sappend. \<Lambda> v ys. Fixrec.run\<cdot>(match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>ys) +++ match_scons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(x :# sappend\<cdot>xs\<cdot>ys)))"
Nset_nset: fixes n :: "nat"   and m :: "nat" shows "{i. i \<le> Suc (n + m)} = {i. i \<le> n} \<union> nset (Suc n) (Suc (n + m))"
ndepth_Push_Node: fixes i :: "nat"   and n :: "('a, 'b) node" shows "ndepth (Push_Node (Inr (Suc i)) n) = Suc (ndepth n)"
state_rank_sink: fixes q :: "'b"   and n :: "nat" assumes "sink q" shows "state_rank q n = None"
map_of_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "('c \<times> 'e) list" shows "table_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> table_of xs"
diff_card_le_card_Diff: fixes B :: "'b set"   and A :: "'b set" assumes "finite B" shows "order.greater_eq (card (A - B)) (card A - card B)"
union_left_cancel: fixes K :: "'c multiset"   and M :: "'c multiset"   and N :: "'c multiset" shows "(K + M = K + N) = (M = N)"
f': shows "has_bochner_integral (lebesgue_on {- b..- a}) (\<lambda>x. f (- x)) i"
card_bind: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c set" assumes "finite S"   and "\<forall>X\<in>S. finite (f X)"   and "disjoint_family_on f S" shows "card (S \<bind> f) = (\<Sum>x\<in>S. card (f x))"
cert_stepsize_impl_nres_def: fixes E :: "nat"   and odoi :: "ode_ops \<times> (floatarith list \<times> floatarith list \<times> floatarith list \<times> floatarith list \<times> nat) \<times> floatarith list list option \<times> (floatarith list \<times> floatarith list \<times> floatarith list \<times> floatarith list \<times> nat) option"   and mi :: "'b list \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'b list \<Rightarrow> ('b list \<times> 'c) option nres"   and X0i :: "'b list"   and hi :: "real"   and ni :: "nat"   and i_i :: "nat" shows "cert_stepsize_impl_nres E odoi mi X0i hi ni i_i \<equiv> rec_nat (\<lambda>((x, xa), xb). let y' = tracing_fun optns ''cert_stepsize failed'' (Some x) in SUCCEED) (\<lambda>x xa ((xb, xc), xd). RETURN (ivl_rep_of_set_impl xb) \<bind> (\<lambda>(xe, xf). nres_of (mk_safe_impl odoi (appr_of_ivl ops xe xf)) \<bind> (\<lambda>xg. nres_of (P_iter_impl E odoi xb xc xd xg) \<bind> case_option (xa ((xb, xc / 2), xd)) (\<lambda>xh. mi xb xc xc xh \<bind> (\<lambda>xi. mi xb 0 xc xh \<bind> (\<lambda>xj. case (xi, xj) of (None, xl) \<Rightarrow> let y' = tracing_fun optns ''cert_stepsize method failed'' (Some xh) in xa ((xb, xc / 2), xd) | (Some (xm, xn), None) \<Rightarrow> let y' = tracing_fun optns ''cert_stepsize method failed'' (Some xh) in xa ((xb, xc / 2), xd) | (Some (xm, xn), Some (xo, xp)) \<Rightarrow> nres_of (mk_safe_impl odoi xm) \<bind> (\<lambda>xq. nres_of (mk_safe_impl odoi xo) \<bind> (\<lambda>xr. RETURN (xc, xm, xo, xn))))))))) i_i ((X0i, hi), ni)"
offset_poly_eq_0_iff: fixes p :: "'a poly"   and h :: "'a" shows "(offset_poly p h = 0) = (p = 0)"
sbintrunc_Suc_numeral_2: fixes n :: "nat"   and w :: "num" shows "signed_take_bit (Suc n) (numeral (num.Bit0 w)) = 2 * signed_take_bit n (numeral w)"
simps_17: fixes f1 :: "name \<Rightarrow> 'a"   and f2 :: "name \<Rightarrow> 'a"   and f3 :: "(Term_Class.term \<times> sterm) list \<Rightarrow> 'a"   and f4 :: "sterm \<Rightarrow> sterm \<Rightarrow> 'a"   and x1 :: "name" shows "(case Sconst x1 of Sconst x \<Rightarrow> f1 x | Svar x \<Rightarrow> f2 x | Sabs x \<Rightarrow> f3 x | x $\<^sub>s xa \<Rightarrow> f4 x xa) = f1 x1"
fct_gen_subalgebra_sigma_sets: fixes X :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "X \<in> space M \<rightarrow> space N" shows "sets (fct_gen_subalgebra M N X) = {X -` B \<inter> space M |B. B \<in> sets N}"
select_convs_1: fixes name :: "mname"   and parTs :: "ty list"   and more :: "'a" shows "name \<lparr>name = name, parTs = parTs, \<dots> = more\<rparr> = name"
wbalanced_subtrees: fixes t :: "'b tree"   and s :: "'b tree" assumes "wbalanced t"   and "s \<in> subtrees t" shows "wbalanced s"
iMOD_cut_le1: fixes t :: "nat"   and r :: "nat"   and m :: "nat" assumes "t \<in> [ r, mod m ]" shows "[ r, mod m ] \<down>\<le> t = [ r, mod m, (t - r) div m ]"
linear_injective_left_inverse: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" assumes "vector_space_pair s1 s2"   and "linear s1 s2 f"   and "inj f" shows "\<exists>g. linear s2 s1 g \<and> g \<circ> f = id"
comp_fun_commute: fixes y :: "'c"   and x :: "'c" shows "Mapping.update y p \<circ> Mapping.update x p = Mapping.update x p \<circ> Mapping.update y p"
filter_ii: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters"   and "a \<in> F"   and "a \<le> b" shows "b \<in> F"