hcju/isabelleps huggingface dataset:
test_bit_lenD: fixes x :: "'a word"   and n :: "nat" assumes "x !! n" shows "n < LENGTH('a) \<and> x !! n"
assms_2: shows "\<forall>((r, c), uu_)\<in>set ts. r < n"
invar_def: fixes C :: "'a set"   and F :: "'a set set" shows "invar C F = (F \<subseteq> E \<and> vertex_cover (E - F) C \<and> finite C \<and> (\<exists>M. inv_matching C F M))"
pfun_app_empty: fixes x :: "'b" shows "{}\<^sub>p(x)\<^sub>p = undefined"
setToListFresh_4: fixes xvec :: "name list"   and Xs :: "name set" shows "set xvec \<sharp>* Xs = xvec \<sharp>* Xs"
get_root_node_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok Shadow_DOM.get_root_node ptr"
MPair: fixes U :: "freemsg"   and V :: "freemsg" shows "MPair (Abs_Msg (msgrel `` {U})) (Abs_Msg (msgrel `` {V})) = Abs_Msg (msgrel `` {MPAIR U V})"
simps_1: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and at :: "('h, 'i) atm" assumes "Signature TYPE('e) wtFsym wtPsym" shows "fsymsL (Pos at) = fsymsA at"
bp_bins_finite: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and P :: "'b set set" assumes "BinPacking_Complete U w c"   and "BinPacking_Complete.bp U w c P" shows "\<forall>B\<in>P. finite B"
k_mkAlgAuto_implements: shows "Clock.implements Clock.k_mkAlgAuto"
prj_chine_assoc'_3: shows "trg_trg_f.Prj\<^sub>0 \<cdot> trg_trg_f.chine_assoc' = trg_trg_f.Prj\<^sub>0\<^sub>0"
has_integral_erf_aux: shows "((\<lambda>t. exp (- t\<^sup>2)) has_integral sqrt pi / 2) {0..}"
monad_discard_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_discard return bind" shows "monad_discard (return_env return) (bind_env bind)"
sl1: shows "- (ordinate A - ordinate B) \<in> radical_sqrt"
fls_inverse_zero_conv_fls_const: shows "inverse 0 = fls_const (inverse (0::'a))"
cextend_basis_raw_def: shows "cextend_basis \<equiv> vector_space.extend_basis (*\<^sub>C)"
A5: shows "\<forall>a b c d a' b' c' d'. a \<noteq> b \<and> B a b c \<and> B a' b' c' \<and> a b \<equiv> a' b' \<and> b c \<equiv> b' c' \<and> a d \<equiv> a' d' \<and> b d \<equiv> b' d' \<longrightarrow> c d \<equiv> c' d'"
vsingleton_iff: fixes y :: "V"   and x :: "V" shows "(y \<in>\<^sub>\<circ> set {x}) = (y = x)"
inf_sup_distrib1: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" shows "(x \<and> (y \<or> z)) = (x \<and> y \<or> x \<and> z)"
one_closed: shows "\<one>\<^bsub>K [X]\<^esub> \<in> carrier (K [X])"
mult_right_sub_dist_sup_left: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x * z \<le> (x \<squnion> y) * z"
sub_wz: shows "cbox a b \<union> cbox c d \<subseteq> box w z"
assms_5: shows "insertion (nth_default 0 (xs' @ x # xs)) d = D"
graph_union_preserves_restrict: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" assumes "graph G\<^sub>1"   and "graph G\<^sub>2" shows "graph (graph_union G\<^sub>1 G\<^sub>2)"
simps_2: fixes l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "left (tree_iterate l r s) = tree_iterate l r (l s)"
fmap_ext_fmdom: fixes N :: "('c, 'd) fmap"   and N' :: "('c, 'd) fmap" assumes "fmdom N = fmdom N'"   and "\<And>x. x |\<in>| fmdom N \<Longrightarrow> fmlookup N x = fmlookup N' x" shows "N = N'"
left_neutral: fixes a :: "'e" shows "(1::'e) * a = a"
mset_subst_list: fixes x :: "'d"   and t :: "('c, 'd) Term.term"   and ys :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "mset (subst_list (subst x t) ys) = subst_mset (subst x t) (mset ys)"
Src_Src: fixes t :: "'a local.term" assumes "Arr t" shows "Src (Src t) = Src t"
\<tau>_def: fixes t :: "real" shows "\<tau> t = t - \<delta>"
nomulti_digraphI_app_iso: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "nomulti_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "nomulti_digraph (app_iso hom G)"
ge_zero: fixes x :: "'a" assumes "x \<in> V" shows "0 \<le> \<parallel>x\<parallel>"
split: fixes P :: "'h \<Rightarrow> bool"   and f1 :: "'i \<Rightarrow> 'h"   and f2 :: "'j \<Rightarrow> 'h"   and sum :: "'i + 'j" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Inr x2 \<longrightarrow> P (f2 x2)))"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<parallel> (a \<parallel> c) = a \<parallel> (b \<parallel> c)"
IH: fixes x :: "nat" assumes "x < base ^ n" shows "(\<Sum>i<n. x div base ^ i mod base * base ^ i) = x"
k_def: shows "k = \<lfloor>(a - a0) / pp\<rfloor>"
list_all_set: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. P x" shows "list_all P xs"
diff_less_top_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(a - b < top) = (a < top)"
lbvs_axioms: shows "lbvs A r f \<top> \<bottom> step c ins"
I1: shows "set_integrable lborel {0..1 / 2} (\<lambda>t. t powr (a - 1) * (1 - t) powr (b - 1))"
assms_3: shows "eval (Pr n f g) (x # xs) \<down>"
simps_1: fixes pending_write :: "bool"   and A :: "nat set" shows "acquired_reads pending_write [] A = {}"
det3_eq_zero: fixes e :: "real"   and xr :: "real \<times> real"   and Q :: "real \<times> real" assumes "e \<noteq> 0" shows "coll 0 xr (e *\<^sub>R Q) = coll 0 xr Q"
plus_oracle_stop_split: fixes P :: "(('a + 'b) option \<times> 'c) spmf \<Rightarrow> bool"   and left :: "'c \<Rightarrow> 'd \<Rightarrow> ('a option \<times> 'c) spmf"   and right :: "'c \<Rightarrow> 'e \<Rightarrow> ('b option \<times> 'c) spmf"   and s :: "'c"   and lr :: "'d + 'e" shows "P ((left \<oplus>\<^sub>O\<^sup>S right) s lr) = ((\<forall>x. lr = Inl x \<longrightarrow> P (map_spmf (apfst (map_option Inl)) (left s x))) \<and> (\<forall>y. lr = Inr y \<longrightarrow> P (map_spmf (apfst (map_option Inr)) (right s y))))"
ChiStar_ChiPlus_noteq: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "S \<noteq> T"   and "(S, T) \<in> ChiStar A" shows "(S, T) \<in> ChiPlus A"
uint_mod: fixes x :: "'a word"   and y :: "'a word" shows "uint (x mod y) = uint x mod uint y"
integral_def: fixes i :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "integral i f = (SOME y. (f has_integral y) i \<or> \<not> f integrable_on i \<and> y = (0::'a))"
pi': shows "set pi \<subseteq> supp p \<times> pi \<bullet> supp p"
F\<psi>: shows "\<guillemotleft>F \<psi> : F.map\<^sub>0 (G.map\<^sub>0 (src\<^sub>C f)) \<rightarrow>\<^sub>C F.map\<^sub>0 (G.map\<^sub>0 (trg\<^sub>C f))\<guillemotright>"
simps: fixes a :: "'a set \<Rightarrow> 'a set" shows "IK a = (a = id \<or> a = I \<or> a = K \<or> (\<exists>f g. a = f \<circ> g \<and> IK f \<and> IK g))"
return_pmf_inj: fixes x :: "'b"   and y :: "'b" shows "(return_pmf x = return_pmf y) = (x = y)"
lead_coeff_list_def: fixes p :: "'c poly" shows "lead_coeff p = (if coeffs p = [] then 0::'c else last (coeffs p))"
eval_wp_Skip: fixes P :: "'b \<Rightarrow> real" shows "wp Skip P = P"
reachability_closed_under_execution: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "reachable s"   and "s \<leadsto>\<^sup>* s'" shows "reachable s'"
kat_2_equiv_opp: fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "(ad z \<cdot> y \<le> y \<cdot> ad x) = (ad z \<cdot> y \<cdot> d x = (zero_class.zero::'a))"
simps: fixes n :: "nat"   and s :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end n (s, l, r) = (if s = 0 then inv_end0 n (l, r) else if s = 1 then inv_end1 n (l, r) else if s = 2 then inv_end2 n (l, r) else if s = 3 then inv_end3 n (l, r) else if s = 4 then inv_end4 n (l, r) else if s = 5 then inv_end5 n (l, r) else False)"
simps_1: shows "opt_MatchAny_match_expr_once MatchAny = MatchAny"
bdd: shows "bdd_below {x. \<omega> \<le> f x}"
finite_UN: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "\<Union> (A ` I) \<in> sets M"
degree_linear_power: fixes a :: "'c"   and n :: "nat" shows "degree ([:a, 1::'c:] ^ n) = n"
surjective: fixes r :: "('a, 'b, 'c, 'd) stateSP_scheme" shows "r = \<lparr>globals = globals r, locals = locals r, \<dots> = stateSP.more r\<rparr>"
assms: shows "Epigraph UNIV f = Epigraph UNIV g"
zero_neq_neg_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "zero \<noteq> uminus (numeral.numeral one plus n)"
elims: fixes h :: "'c \<Rightarrow> 'b"   and x :: "'c rose_tree"   and y :: "('a, 'b) rose_tree\<^sub>m" assumes "blind_source_tree h x = y" obtains data :: "'c" and subtrees :: "'c rose_tree list" where "x = Tree (data, subtrees)" and "y = Tree\<^sub>m (Blinded (Content (h data, map (hash_source_tree h) subtrees)))"
that: fixes l :: "'a list" assumes "length l = n"   and "\<forall>i<n. P (l ! i) i" shows "thesis"
orderI: fixes a :: "'e"   and b :: "'e" assumes "a = a \<sqinter> b" shows "a \<sqsubseteq> b"
JF2col_0: fixes x :: "'d JF2" shows "JF2col 0 x = {}"
one_times_ivl_left: fixes A :: "'b interval" shows "1 * A = A"
rev_iterateoi_correct: fixes S :: "('a, unit) RBT.rbt" assumes "rs.invar S" shows "set_iterator_rev_linord (rs.reverse_iterateoi S) (rs.\<alpha> S)"
Finite_Markov_Decision_Process_axioms: shows "Finite_Markov_Decision_Process \<K> \<S>"
simps_1: fixes i :: "int" shows "reducecoeffh (C i) = (\<lambda>g. C (i div g))"
higher_derivatives_exists: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and H :: "'b" assumes "higher_differentiable_on X f n"   and "open X" obtains Df :: "'b \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'c" where "\<And>a h. Df a 0 h = f a" and "\<And>a h i. \<lbrakk>i < n; a \<in> X\<rbrakk> \<Longrightarrow> ((\<lambda>a. Df a i H) has_derivative Df a (Suc i)) (at a)" and "\<And>a i. \<lbrakk>i \<le> n; a \<in> X\<rbrakk> \<Longrightarrow> Df a i H = nth_derivative i f a H"
edge_rev_a_neigh_neq: shows "edge_rev M a_neigh \<noteq> a'"
irrefl_def: fixes r :: "('b \<times> 'b) set" shows "irrefl r = (\<forall>a. (a, a) \<notin> r)"
dd2: shows "DD2 \<in> {DD. set_mset DD \<subseteq> N \<and> set_mset DD \<union> CC \<Turnstile> {E} \<and> (\<forall>D'\<in>#DD. D' < D)}"
induct: fixes x1 :: "state"   and x2 :: "com"   and x3 :: "nat"   and x4 :: "state"   and P :: "state \<Rightarrow> com \<Rightarrow> nat \<Rightarrow> state \<Rightarrow> bool" assumes "x1 -x2-x3\<rightarrow> x4"   and "\<And>t f s n. t \<in> f s \<Longrightarrow> P s (Do f) n t"   and "\<And>s0 c0 n s1 c1 s2. \<lbrakk>s0 -c0-n\<rightarrow> s1; P s0 c0 n s1; s1 -c1-n\<rightarrow> s2; P s1 c1 n s2\<rbrakk> \<Longrightarrow> P s0 (c0; c1) n s2"   and "\<And>b s c0 n t c1. \<lbrakk>b s; s -c0-n\<rightarrow> t; P s c0 n t\<rbrakk> \<Longrightarrow> P s (IF b THEN c0 ELSE c1) n t"   and "\<And>b s c1 n t c0. \<lbrakk>\<not> b s; s -c1-n\<rightarrow> t; P s c1 n t\<rbrakk> \<Longrightarrow> P s (IF b THEN c0 ELSE c1) n t"   and "\<And>b s c n. \<not> b s \<Longrightarrow> P s (WHILE b DO c) n s"   and "\<And>b s c n t u. \<lbrakk>b s; s -c-n\<rightarrow> t; P s c n t; t -WHILE b DO c-n\<rightarrow> u; P t (WHILE b DO c) n u\<rbrakk> \<Longrightarrow> P s (WHILE b DO c) n u"   and "\<And>s p n t. \<lbrakk>s -body p-n\<rightarrow> t; P s (body p) n t\<rbrakk> \<Longrightarrow> P s (CALL p) (Suc n) t"   and "\<And>f s c n t g. \<lbrakk>f s -c-n\<rightarrow> t; P (f s) c n t\<rbrakk> \<Longrightarrow> P s (LOCAL f; c; g) n (g s t)" shows "P x1 x2 x3 x4"
capacity_outside: fixes \<Delta> :: "('a, 'b) network_scheme"   and e :: "'a \<times> 'a" assumes "countable_network \<Delta>"   and "e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub>" shows "capacity \<Delta> e = 0"
redT_updW_None_SomeD: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and t' :: "'b"   and w :: "'c wait_set_status" assumes "redT_updW t ws wa ws'"   and "ws' t' = \<lfloor>w\<rfloor>"   and "ws t' = None" shows "t = t' \<and> (\<exists>w'. w = InWS w' \<and> wa = Suspend w')"
set_le_in: fixes n :: "'a"   and I :: "'a set" shows "{a. a < n \<and> a \<in> I} = {a \<in> I. a < n}"
eSum_tr: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and l :: "nat"   and g :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. x j \<in> carrier K) \<and> (\<forall>j\<le>n. b j \<in> carrier K) \<and> l \<le> n \<and> (\<forall>j\<in>{h. h \<le> n} - {l}. g j = x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a b j)) \<and> g l = x l \<cdot>\<^sub>r (-\<^sub>a b l) \<longrightarrow> \<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a b j)) n \<plusminus> -\<^sub>a x l = \<Sigma>\<^sub>e K g n"
connected_component_of_subset: fixes S :: "'c set"   and x :: "'c"   and y :: "'c"   and T :: "'c set" assumes "connected_component S x y"   and "S \<subseteq> T" shows "connected_component T x y"
Sbis_Sym: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>s d" shows "d \<approx>s c"
weakly_Pareto_optimal_for_men_def: fixes \<mu> :: "('m \<times> 'w) set" shows "weakly_Pareto_optimal_for_men \<mu> = (simple \<mu> \<and> (\<nexists>\<mu>''. simple \<mu>'' \<and> weakly_preferred_by_men \<mu> \<mu>'' \<and> strictly_preferred_by_a_man \<mu> \<mu>''))"
linear: fixes f :: "'a \<Rightarrow> 'b" assumes "clinear f" shows "linear f"
align\<^sub>d_K: shows "local.aligned\<^sub>d e K"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[], s\<rangle> \<Rightarrow>\<^sub>s t"
linorder_prod: shows "class.linorder (less_eq_prod (\<sqsubseteq>\<^sub>a) (\<sqsubset>\<^sub>a) (\<sqsubseteq>\<^sub>b)) (less_prod (\<sqsubseteq>\<^sub>a) (\<sqsubset>\<^sub>a) (\<sqsubset>\<^sub>b))"
sorted_transpose: fixes xs :: "'g list list" shows "sorted (rev (map length (transpose xs)))"
simplicial_simplex: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "simplicial_simplex p S f = (f ` standard_simplex p \<subseteq> S \<and> (\<exists>l. f = oriented_simplex p l))"
many_strongly_connected_implies_8: fixes x :: "'a" assumes "many_strongly_connected x" shows "x ; x\<^sup>T \<le> x\<^sup>+"
eval_fm_Conj: fixes e :: "(name, hf) finfun"   and A :: "fm"   and B :: "fm" shows "eval_fm e (A AND B) = (eval_fm e A \<and> eval_fm e B)"
assms_16: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
typeof_f32: fixes v :: "v" assumes "typeof v = T_f32" shows "\<exists>c. v = ConstFloat32 c"
power2_nat_le_eq_le: fixes m :: "nat"   and n :: "nat" shows "(m\<^sup>2 \<le> n\<^sup>2) = (m \<le> n)"
mems_le_finite: fixes s :: "nat set"   and k :: "nat" assumes "\<forall>x. x \<in> s \<longrightarrow> x \<le> k" shows "finite s"
Un_ac_3: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = B \<union> A"
Par2F: fixes Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto> \<alpha> \<prec> Q'" shows "P \<parallel> Q \<longmapsto> \<alpha> \<prec> P \<parallel> Q'"
lead_coeff_not_zero: fixes K :: "'a set"   and a :: "'a"   and p :: "'a list" assumes "polynomial K (a # p)" shows "a \<in> K - {\<zero>}"
l1C1': shows "l\<^sub>1 \<in> C\<^sub>1'"
p_t_stop: shows "ego.p ego2.t_stop = ego.p_max"
power_strict_increasing: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "nat"   and N :: "nat"   and a :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n < N"   and "less one a" shows "less (power.power one times a n) (power.power one times a N)"
bisim_must_sync_preserve1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and t :: "'b"   and x1 :: "'a"   and m1 :: "'c"   and x2 :: "'g"   and m2 :: "'h" assumes "FWbisimulation final1 r1 final2 r2 bisim"   and "bisim t (x1, m1) (x2, m2)"   and "multithreaded_base.must_sync final1 r1 t x1 m1" shows "multithreaded_base.must_sync final2 r2 t x2 m2"
invertible_left_inverse: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "Group_Theory.monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M" shows "composition (monoid.inverse M composition unit u) u = unit"
sets: shows "\<Union> (N ` f ` space M) \<in> null_sets M"
unitK_def: shows "unitK \<equiv> \<Lambda> a c. c\<cdot>a"
star_one_sup_below: fixes x :: "'a"   and z :: "'a" shows "x \<sqinter> top \<sqinter> (top \<squnion> z) \<le> x \<sqinter> top"
wf_decr: fixes k :: "'k"   and idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and l :: "nat" assumes "local.wf (SUC k idx) \<phi>"   and "LEQ k l idx"   and "\<not> local.find k l \<phi>" shows "local.wf idx (decr k l \<phi>)"
is_Nil_def: fixes a :: "'b list" shows "is_Nil a \<equiv> case a of [] \<Rightarrow> True | a # list \<Rightarrow> False"
int_not_def: fixes k :: "int" shows "NOT k = - k - 1"
addpre_simps_1: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and pre :: "nat set" shows "\<pi>\<^sub>2 (addpre v pre) = \<pi>\<^sub>2 v"
stake_suffix_id: fixes i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "stake i \<sigma> @- (\<sigma> |\<^sub>s i) = \<sigma>"
real_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and real_of :: "'a \<Rightarrow> real" assumes "class.real_embedding plus zero minus uminus less_eq less sgn abs times one real_of" shows "real_of zero = 0"
Pd_shuffle_to_top_def: fixes ds' :: "'d set"   and Y :: "'x set" shows "Pd_shuffle_to_top ds' Y = override_on Pd (shuffle_to_top Y) ds'"
limit_nonempty: fixes x :: "nat \<Rightarrow> 'b" assumes "finite (range x)" shows "\<exists>a. a \<in> limit x"
poly_rel_degree: shows "(poly_rel ===> (=)) degree_i degree"
h: shows "\<Prod>(insert x F) = x * \<Prod>F"
a4: shows "\<Squnion> (L2_set (cmod \<circ> (\<lambda>i. x i + y i)) ` Collect finite) \<le> (SUP xa\<in>Collect finite. L2_set (cmod \<circ> x) xa + L2_set (cmod \<circ> y) xa)"
\<section>: fixes \<gamma> :: "'a \<Rightarrow> 'a set"   and e :: "real" assumes "gauge \<gamma>"   and "\<And>f \<D>. \<lbrakk>f \<in> F; \<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R f x) - integral (cbox a b) f) < e" shows "\<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>f \<D>. f \<in> (\<lambda>f. f \<circ> uminus) ` F \<and> \<D> tagged_division_of cbox (- b) (- a) \<and> \<gamma> fine \<D> \<longrightarrow> norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R f x) - integral (cbox (- b) (- a)) f) < e)"
test_def: fixes x :: "'a option"   and y :: "'b list" shows "test x y = (case x of None \<Rightarrow> case y of [] \<Rightarrow> 1::'a | x # xa \<Rightarrow> 2::'a | Some x \<Rightarrow> x)"
vwalk_to_vpath_last: fixes xs :: "'a list" shows "last (vwalk_to_vpath xs) = last xs"
prems_2: shows "\<langle>while (b) c',s1\<rangle> \<rightarrow>* \<langle>Skip,s2\<rangle>"
inter_guards_DynCom: fixes f1 :: "'d \<Rightarrow> ('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((DynCom f1 \<inter>\<^sub>g c2) = Some c) = (\<exists>f2. c2 = DynCom f2 \<and> (\<forall>s. (f1 s \<inter>\<^sub>g f2 s) \<noteq> None) \<and> c = DynCom (\<lambda>s. the (f1 s \<inter>\<^sub>g f2 s)))"
upd_conv_take_nth_drop: fixes i :: "nat"   and xs :: "'d list"   and a :: "'d" assumes "i < length xs" shows "xs[i := a] = take i xs @ a # drop (Suc i) xs"
continuous: fixes f :: "'d \<Rightarrow> 'e"   and F :: "'f filter"   and g :: "'f \<Rightarrow> 'd" assumes "bounded_linear f"   and "continuous F g" shows "continuous F (\<lambda>x. f (g x))"
pol_deg_n: fixes p :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat" assumes "p \<in> carrier R"   and "pol_coeff S c"   and "n \<le> fst c"   and "p = polyn_expr R X n c"   and "snd c n \<noteq> \<zero>\<^bsub>S\<^esub>" shows "deg_n R S X p = n"
div_greater_zero_iff: fixes m :: "nat"   and n :: "nat" shows "(0 < m div n) = (n \<le> m \<and> 0 < n)"
prefixes_def: fixes j :: "nat"   and t :: "nat" shows "local.prefixes j t \<equiv> the (eval r_prefixes [t, j])"
coeff_a_inv: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)" shows "coeff (UP R) (\<ominus>\<^bsub>UP R\<^esub> p) n = \<ominus>\<^bsub>R\<^esub> coeff (UP R) p n"
is_iso_def: fixes G :: "('f, 'g) monoid_scheme"   and H :: "('h, 'i) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
ArityAnalysisSafe_axioms: shows "ArityAnalysisSafe Aexp"
BuD_def: shows "BuD = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0<..int q - 1}}"
reachable_alt_def: fixes A :: "('e, 'f, 'd, 'g) FSM_scheme"   and p :: "'d" shows "reachable A p = {target r p |r. path A r p}"
Nats_mult_nonpos_Ints: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<nat>"   and "y \<in> \<int>\<^sub>\<le>\<^sub>0" shows "x * y \<in> \<int>\<^sub>\<le>\<^sub>0"
mono_derived_set: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H" shows "derived_set G K \<subseteq> derived_set G H"
abs_minus_cancel: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (uminus a) = abs a"
simps: fixes xs :: "'a list" shows "permutations_of_list_impl xs = (if xs = [] then [[]] else List.bind (remdups xs) (\<lambda>x. map ((#) x) (permutations_of_list_impl (remove1 x xs))))"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "a \<in> set1_FGco x"   and "b \<in> set1_FGco y"   and "L1 a b" shows "L1' a b"
norm_not_less_zero: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<not> norm x < 0"
swapWatchesEffect_2: fixes clause :: "nat"   and state :: "State"   and clause' :: "nat" shows "getWatch2 (swapWatches clause state) clause' = (if clause = clause' then getWatch1 state clause' else getWatch2 state clause')"