hcju/isabelleps huggingface dataset:
subset: shows "set (map fst ps0) \<subseteq> \<V> g"
to_woot_character_insert_no_eff: fixes m :: "('a, 'b) insert_message"   and M :: "('a, 'b) message set" shows "to_woot_character (insert (Insert m) M) = to_woot_character M"
e: shows "euclid_ext2 a b = (p, q, u, v, d)"
dbm_entry_val_mono_1: fixes u :: "'b \<Rightarrow> 'c"   and c :: "'b"   and c' :: "'b"   and b :: "'c DBMEntry"   and b' :: "'c DBMEntry" assumes "dbm_entry_val u (Some c) (Some c') b"   and "b \<preceq> b'" shows "dbm_entry_val u (Some c) (Some c') b'"
lessThan_empty_iff: fixes n :: "nat" shows "({..<n} = {}) = (n = 0)"
geq': fixes x :: "real" assumes "x \<in> {0..t}" shows "g' x s \<le> f' x s"
isBoolV_def: fixes v :: "Value" shows "isBoolV v = (case v of boolV b \<Rightarrow> True | _ \<Rightarrow> False)"
concurrent_ops_commute_rearrange: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" shows "concurrent_ops_commute (xs @ x # ys) = concurrent_ops_commute (xs @ ys @ [x])"
\<gamma>1's: shows "set (labels \<gamma>1') \<subseteq> r \<down>s {fst \<beta>_step}"
erfc_def: fixes z :: "'a" shows "erfc z = (1::'a) - erf z"
iso_iff: shows "(hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id \<in> Group.iso (relative_homology_group (int n) (lsphere n) (equator n)) (relative_homology_group (int n) (nsphere n) (upper n))) = (carrier (relative_homology_group (int n) (nsphere n) (upper n)) \<subseteq> hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id ` carrier (relative_homology_group (int n) (lsphere n) (equator n)) \<and> (\<forall>x\<in>carrier (relative_homology_group (int n) (lsphere n) (equator n)). hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id x = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (upper n)\<^esub> \<longrightarrow> x = \<one>\<^bsub>relative_homology_group (int n) (lsphere n) (equator n)\<^esub>))"
\<Phi>_simp_1: fixes y :: "'d"   and x :: "'c" assumes "D.ide y"   and "C.ide x" shows "S.arr (\<Phi> (y, x))"
succ_set_on_bound_2: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and dir :: "('i, 'a) Direction" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>i \<in> rvars (\<T> s')"   and "dir = Positive \<or> dir = Negative" shows "\<langle>\<V> s'\<rangle> x\<^sub>i = the (\<B>\<^sub>l s x\<^sub>i) \<or> \<langle>\<V> s'\<rangle> x\<^sub>i = the (\<B>\<^sub>u s x\<^sub>i)"
PiE_mem: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "f \<in> Pi\<^sub>E S T"   and "x \<in> S" shows "f x \<in> T x"
Rep_unit_inverse: fixes x :: "Linear_Aggregation_Algebras.unit" shows "Linear_Aggregation_Algebras.unit.Abs_unit (Linear_Aggregation_Algebras.unit.Rep_unit x) = x"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and xa :: "'b \<Rightarrow>\<^sub>0 'c" shows "poly_mapping.lookup (x + xa) = (\<lambda>k. poly_mapping.lookup x k + poly_mapping.lookup xa k)"
inf_img_fin_dom: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "finite (f ` A)"   and "infinite A" shows "\<exists>y\<in>f ` A. infinite (f -` {y})"
intro: fixes Pd :: "'d \<Rightarrow> ('e \<times> 'e) set"   and Xd :: "'e \<Rightarrow> 'd"   and Ch :: "'f \<Rightarrow> 'e set \<Rightarrow> 'e set"   and Xh :: "'e \<Rightarrow> 'f" assumes "\<forall>d. Linear_order (Pd d)"   and "\<forall>d. Field (Pd d) \<subseteq> {x. Xd x = d}"   and "\<forall>h X. Ch h X \<subseteq> {x \<in> X. Xh x = h}"   and "\<forall>h X. inj_on Xd (Ch h X)" shows "Contracts Xd Xh Pd Ch"
generated_ring_INTER: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring" shows "\<Inter> (A ` I) \<in> generated_ring"
boundedI: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> inf b c"
cases: fixes rest1 :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme"   and rest2 :: "('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'h spmf"   and a1 :: "'a spmf"   and a2 :: "'h spmf" assumes "trace_rest_closure rest1 rest2 IA IU p q a1 a2" obtains tr :: "('c \<times> 'e \<times> 'b list + 'd \<times> 'f \<times> 'b list) list" where "a1 = trace_rest_aux rest1 p tr" and "a2 = trace_rest_aux rest2 q tr" and "set tr \<subseteq> IA \<times> UNIV <+> IU \<times> UNIV"
arr_dom: fixes f :: "'a1" assumes "A1.arr f" shows "A1.arr (A1.dom f)"
independent_UnI: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" assumes "vector_space scale"   and "\<not> module.dependent scale A"   and "\<not> module.dependent scale B"   and "module.span scale A \<inter> module.span scale B = {0::'c}" shows "\<not> module.dependent scale (A \<union> B)"
cobounded2: fixes b :: "'a upred"   and a :: "'a upred" shows "(a \<or> b) \<sqsubseteq> b"
pexp_Suc: fixes p :: "'a list"   and n :: "nat" shows "p %^ Suc n = p *** p %^ n"
bounded_iff: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and c :: "'b \<times> 'c" shows "less_eq_prod a (inf_prod b c) = (less_eq_prod a b \<and> less_eq_prod a c)"
induct: fixes x1 :: "type_scheme list"   and x2 :: "expr"   and x3 :: "typ"   and P :: "type_scheme list \<Rightarrow> expr \<Rightarrow> typ \<Rightarrow> bool" assumes "x1 |- x2 :: x3"   and "\<And>n A t. \<lbrakk>n < length A; t <| A ! n\<rbrakk> \<Longrightarrow> P A (Var n) t"   and "\<And>t1 A e t2. \<lbrakk>mk_scheme t1 # A |- e :: t2; P (mk_scheme t1 # A) e t2\<rbrakk> \<Longrightarrow> P A (Abs e) (t1 -> t2)"   and "\<And>A e1 t2 t1 e2. \<lbrakk>A |- e1 :: t2 -> t1; P A e1 (t2 -> t1); A |- e2 :: t2; P A e2 t2\<rbrakk> \<Longrightarrow> P A (App e1 e2) t1"   and "\<And>A e1 t1 e2 t. \<lbrakk>A |- e1 :: t1; P A e1 t1; gen A t1 # A |- e2 :: t; P (gen A t1 # A) e2 t\<rbrakk> \<Longrightarrow> P A (LET e1 e2) t" shows "P x1 x2 x3"
sbelow_rel_def: fixes A :: "('a \<times> 'c) set" shows "\<langle>A\<rangle>sbelow_rel = \<langle>A\<rangle>sctn_rel O br sbelow_halfspace top"
BIT_special_simps_2: shows "0 BIT True = 1"
pequiv_comm: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<simeq> b = b \<simeq> a"
in_interval_to_interval: fixes a :: "'b" shows "a \<in>\<^sub>i interval_of a"
MOST_I: fixes P :: "'a \<Rightarrow> bool" assumes "\<And>x. P x" shows "\<forall>\<^sub>\<infinity>x. P x"
shadow_root_ptr_kinds_eq: shows "shadow_root_ptr_kinds h = shadow_root_ptr_kinds h'"
derive_size_2: fixes x :: "'b sort_key extended \<times> 'b sort_key extended"   and i :: "'b" assumes "snd x \<le> \<lbrakk>Final i\<rbrakk> \<and> is_interval x" shows "int_size (derive_right x) < int_size x"
tendsto_minus_cancel: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" assumes "((\<lambda>x. - f x) \<longlongrightarrow> - a) F" shows "(f \<longlongrightarrow> a) F"
eqvt: fixes x1 :: "(Syntax.var, ty) fmap"   and x2 :: "Syntax.term"   and x3 :: "ty"   and p :: "perm" assumes "x1 \<turnstile>\<^sub>W x2 : x3" shows "(p \<bullet> x1) \<turnstile>\<^sub>W p \<bullet> x2 : (p \<bullet> x3)"
alternating_order2_even_cancel_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and s :: "'b"   and t :: "'b"   and n :: "nat"   and m :: "nat" assumes "class.monoid_add plus zero"   and "plus s s = zero"   and "plus t t = zero"   and "gcd_nat.greater_eq n 2"   and "order.greater_eq n m" shows "monoid_add.sum_list plus zero (alternating_list n s t @ alternating_list m t s) = monoid_add.sum_list plus zero (alternating_list (n - m) s t)"
swapEnv_involutive2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "goodEnv rho" shows "rho &[x \<and> y]_zs &[y \<and> x]_zs = rho"
simps_8: fixes x11 :: "'a"   and x12 :: "'b"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x11 \<leftarrow> x12 \<noteq> x31@[x32]"
app_eq1: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "zs = xs @ ys" shows "app xs ys zs"
basic_identity\<^sub>0_def: shows "(\<^bold>=\<^sub>0) \<equiv> \<lambda>F G. (\<^bold>\<lambda>y. F) \<^bold>=\<^sub>1 (\<^bold>\<lambda>y. G)"
Keys_full_gb: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "Keys (fst ` set (full_gb bs)) = (\<lambda>v. term_of_pair (0::'a, component_of_term v)) ` Keys (fst ` set bs)"
infsetsum_set_nn_integral_reals: fixes f :: "nat \<Rightarrow> real" assumes "f abs_summable_on UNIV"   and "\<And>n. 0 \<le> f n" shows "ennreal (infsetsum f UNIV) = \<integral>\<^sup>+x\<in>{0..}. ennreal (f (nat \<lfloor>x\<rfloor>))\<partial>lborel"
cases: fixes r :: "'c ring" obtains carrier :: "'c set" and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" and one :: "'c" and zero :: "'c" and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" where "r = \<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add\<rparr>"
subintervals_1: shows "fst (split_float_interval ivl) \<le> ivl"
ide_is_iso: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "C.VV.iso a"
lemma_2_5_16: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a l\<rightarrow> b \<le> (b l\<rightarrow> c) r\<rightarrow> a l\<rightarrow> c"
B: shows "Gromov_extension f \<circ> Gromov_extension (inv f) = id"
image_mset_add_mset: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and M :: "'d multiset" shows "image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"
ttree_restr_both: fixes S :: "'b set"   and t :: "'b ttree"   and t' :: "'b ttree" shows "ttree_restr S (t \<otimes>\<otimes> t') = ttree_restr S t \<otimes>\<otimes> ttree_restr S t'"
someone_inserted_id: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and k :: "'a"   and v :: "'b"   and f :: "bool"   and n :: "'a option"   and j :: "nat"   and xs1 :: "('a \<times> 'b \<times> bool) list"   and xs2 :: "('a \<times> 'b \<times> bool) list"   and a :: "'a" assumes "es @ [Deliver (i, Insert (k, v, f) n)] prefix of j"   and "apply_operations es = Some xs1"   and "apply_operations (es @ [Deliver (i, Insert (k, v, f) n)]) = Some xs2"   and "a \<in> element_ids xs2"   and "a \<noteq> k" shows "a \<in> element_ids xs1"
scopeFresh: fixes a :: "name"   and P :: "pi" assumes "a \<sharp> P" shows "<\<nu>a>P \<simeq> P"
fresh_star_set: fixes as :: "atom set"   and xs :: "'b list" shows "as \<sharp>* set xs = as \<sharp>* xs"
elims_1: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and y :: "bool" assumes "SkewBinomialHeapStruc.rank_skew_invar x = y" obtains "x = []" and "y"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" where "x = [t__]" and "y"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # t' # bq" and "y = (SkewBinomialHeapStruc.rank t \<le> SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq))"
safe_reach_ff: shows "safe_reach_upto n safe_free_flowing c\<^sub>0"
steps_in_range: fixes tp :: "cell list \<times> cell list"   and A :: "(action \<times> nat) list"   and stp :: "nat" assumes "\<not> is_final (steps0 (1, tp) A stp)"   and "tm_wf0 A" shows "fst (steps0 (1, tp) A stp) \<le> length A div 2"
ex_head_tail: fixes t :: "dB" shows "\<exists>ts h. t = h \<degree>\<degree> ts \<and> ((\<exists>n. h = Var n) \<or> (\<exists>u. h = Abs u))"
minE: fixes \<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and min :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c) option"   and s :: "'a"   and P :: "'b \<times> 'c \<Rightarrow> bool" assumes "map_min \<alpha> invar min"   and "invar s"   and "rel_of (\<alpha> s) P \<noteq> {}" obtains k :: "'b" and v :: "'c" where "min s P = Some (k, v)" and "(k, v) \<in> rel_of (\<alpha> s) P" and "\<forall>(k', v')\<in>rel_of (\<alpha> s) P. k \<le> k'"
Sura_Bura_clopen_subset: fixes S :: "'a set"   and C :: "'a set"   and U :: "'a set" assumes "locally compact S"   and "C \<in> components S"   and "compact C"   and "open U"   and "C \<subseteq> U" obtains K :: "'a set" where "openin (top_of_set S) K" and "compact K" and "C \<subseteq> K" and "K \<subseteq> U"
concatMap_def: shows "concatMap = (\<Lambda> f. concat oo map\<cdot>f)"
simps_2: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
combine_t_wise_balance: shows "t_wise_balance \<V>\<^sup>+ \<B>\<^sup>+ \<t> (\<Lambda>\<^sub>t + \<Lambda>\<^sub>t')"
rtrancl_idemp: fixes r :: "('b \<times> 'b) set" shows "(r\<^sup>*)\<^sup>* = r\<^sup>*"
rel_dds_conversep: fixes A :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_dds A\<inverse>\<inverse> B\<inverse>\<inverse> = (rel_dds A B)\<inverse>\<inverse>"
ideD_1: fixes a :: "'t" assumes "S'.ide a" shows "S'.arr a"
choice: fixes Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<forall>x. \<exists>y. Q x y" shows "\<exists>f. \<forall>x. Q x (f x)"
D_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "linordered_ab_monoid_add.D plus less_eq m i j k \<equiv> linorder.Min less_eq {linordered_ab_monoid_add.len plus m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
Var_NF: fixes n :: "nat" shows "NF (dB.Var n)"
Sup_fin_def: shows "Sup_fin \<equiv> semilattice_set.F sup"
remzeroeq: fixes a :: "bit list" shows "bv_to_nat a = bv_to_nat (remzero a)"
akra_bazzi_termI: fixes e :: "real"   and h :: "nat \<Rightarrow> real"   and x\<^sub>1 :: "nat"   and x\<^sub>0 :: "nat"   and t :: "nat \<Rightarrow> nat"   and b :: "real" assumes "0 < e"   and "h \<in> O(\<lambda>x. real x / ln (real x) powr (1 + e))"   and "\<And>x. x\<^sub>1 \<le> x \<Longrightarrow> x\<^sub>0 \<le> t x"   and "\<And>x. x\<^sub>1 \<le> x \<Longrightarrow> t x < x"   and "\<And>x. x\<^sub>1 \<le> x \<Longrightarrow> b * real x + h x = real (t x)" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 b t"
minus_Int_eq: fixes T :: "'a set"   and U :: "'a set" shows "T - (T - U) = T \<inter> U"
H1K_def: shows "H1K = H1 \<inter> K"
r_coset_eq_set_mult: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and x :: "'e" shows "H #>\<^bsub>G\<^esub> x = H <#>\<^bsub>G\<^esub> {x}"
t_def2: fixes seq :: "'observation list"   and obs :: "'observation" shows "t seq obs = card {i. i < length seq \<and> seq ! i = obs}"
quad_def: fixes g :: "graph"   and v :: "nat" shows "quad g v \<equiv> |filter (\<lambda>f. final f \<and> |vertices f| = 4) (facesAt g v)|"
quorum_of_def: fixes p :: "'node"   and Q :: "'node set" shows "quorum_of p Q \<equiv> quorum Q \<and> (p \<notin> W \<or> (\<exists>Sl\<in>slices p. Sl \<subseteq> Q))"
subgroup_in_rcosets: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme" assumes "subgroup H G"   and "Group.group G" shows "H \<in> rcosets\<^bsub>G\<^esub> H"
cone_def: fixes Vs :: "'a vec set" shows "n.cone Vs = {x. \<exists>Ws. finite Ws \<and> Ws \<subseteq> Vs \<and> x \<in> n.finite_cone Ws}"
nMTeqSet: fixes p :: "'b list"   and s :: "'b list"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "set p = set s" shows "none_MT_rules C p = none_MT_rules C s"
parallelE: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<parallel> ys" obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs"
simps_3: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and y41 :: "('a, 'b, 'c) Stmt"   and y42 :: "('a, 'b, 'c) Stmt" shows "(x41 ;; x42 = y41 ;; y42) = (x41 = y41 \<and> x42 = y42)"
filter_tab_all_True_Some: fixes t :: "'a \<Rightarrow> 'b option"   and p :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'a"   and v :: "'b" assumes "\<forall>k y. t k = Some y \<longrightarrow> p k y"   and "t k = Some v" shows "filter_tab p t k = Some v"
Abs_Int_def: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" shows "Abs_Int num' plus' \<gamma> \<equiv> Gamma num' plus' \<gamma>"
top_join: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<top> \<squnion> x .= \<top>"
length_le: fixes y :: "'a list" assumes "y \<in> P" shows "length y \<le> n"
ltl_lmap: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist" shows "ltl (lmap f xs) = lmap f (ltl xs)"
PO_refines_implies_R_image_invariant: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "PO_refines R Ta Tc" shows "reach Tc \<subseteq> R `` reach Ta"
row_def: fixes i :: "'c"   and A :: "(('a, 'b) vec, 'c) vec" shows "row i A = (\<chi>j. A $ i $ j)"
OclAnd_true1: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(true and X) = X"
T_merge_min_height: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree" assumes "ltree l"   and "ltree r" shows "T_merge l r \<le> min_height l + min_height r + 1"
iter_wf: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list" assumes "wf_tr M T" shows "wf_tr M (snd (iter M T))"
\<D>_6: shows "\<Union> \<D> = cbox a b"
le_sup_iff: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(x + y \<le> z) = (x \<le> z \<and> y \<le> z)"
times_divide_eq_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "c \<noteq> \<zero>" shows "a \<otimes> (b \<oslash> c) = a \<otimes> b \<oslash> c"
branch_is_path: fixes ds :: "bool list"   and T :: "tree" assumes "branch ds T" shows "path ds T"
IH: fixes w :: "'a"   and c :: "'a list" assumes "path E w ps x"   and "v \<in> set ps"   and "takeWhile ((\<noteq>) v) ps = c" shows "path E w c v"
FDOM_eff_subset_prob_dom: fixes p :: "('d, 'e) fmap"   and e :: "('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "(p, e) \<in> PROB" shows "fmdom' e \<subseteq> prob_dom PROB"
sum_multiset_singleton: fixes A :: "'k set" shows "(\<Sum>n\<in>A. {#n#}) = mset_set A"
less_TC_trans: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<sqsubset> y"   and "y \<sqsubset> z" shows "x \<sqsubset> z"
choice: fixes Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<forall>x. \<exists>y. Q x y" shows "\<exists>f. \<forall>x. Q x (f x)"
sqrt_nat_ceiling_def: fixes x :: "nat" shows "sqrt_nat_ceiling x = root_nat_ceiling 2 x"
cong_four_block_mat: fixes A1 :: "'b mat"   and B1 :: "'b mat"   and A2 :: "'b mat"   and B2 :: "'b mat"   and A3 :: "'b mat"   and B3 :: "'b mat"   and A4 :: "'b mat"   and B4 :: "'b mat" assumes "A1 = B1"   and "A2 = B2"   and "A3 = B3"   and "A4 = B4" shows "four_block_mat A1 A2 A3 A4 = four_block_mat B1 B2 B3 B4"
blleft: shows "bl \<in># remove1_mset bl \<B>"
graph_nodes_it_def: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool"   and nodes_list_it :: "'d \<Rightarrow> ('e list \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'e list \<Rightarrow> 'e list) \<Rightarrow> 'e list \<Rightarrow> 'e list" shows "graph_nodes_it \<alpha> invar nodes_list_it \<equiv> graph \<alpha> invar \<and> graph_nodes_it_axioms \<alpha> invar nodes_list_it"
T3E: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier"   and "A closed"   and "x \<in> carrier"   and "x \<notin> A" obtains B :: "'a set" and U :: "'a set" where "A \<subseteq> B" and "B open" and "U \<in> nhds x" and "B \<inter> U = {}"
fresh_star_list_cons_1: fixes xvec :: "name list"   and x :: "'d"   and xs :: "'d list" shows "xvec \<sharp>* (x # xs) = (xvec \<sharp>* x \<and> xvec \<sharp>* xs)"
bs1: fixes t :: "lam" shows " t \<longrightarrow>\<^sub>\<beta>\<^sup>* t"
FreeGroup_add_closed: fixes g :: "'a freeword"   and S :: "'a set"   and h :: "'a freeword" assumes "g \<in> FreeGroup S"   and "h \<in> FreeGroup S" shows "g + h \<in> FreeGroup S"
prime_prime_factor: fixes n :: "nat" shows "prime n = (n \<noteq> 1 \<and> (\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n))"
Cons_1: fixes a :: "QDelta atom" assumes "a \<in> snd ` set (Atoms (preprocess' t start))"   and "\<forall>var\<in>vars_constraints (map snd t). var < start" shows "atom_var a < FirstFreshVariable (preprocess' t start)"
progress_L0: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and cs :: "e list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "const_list cs" shows "\<lparr>s;vs;cs @ es @ []\<rparr> \<leadsto>_ i \<lparr>s';vs';cs @ es' @ []\<rparr>"
convex_halfspace_lt: fixes a :: "'a"   and b :: "real" shows "convex {x. a \<bullet> x < b}"
supp_map: fixes ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>" shows "supp (map_transform (\<lambda>x0 x1. transform_sumC (x0, x1)) ae as) \<subseteq> supp as"
zero_order_2: fixes n :: "'a multiset" shows "(n \<subseteq># {#}) = (n = {#})"
simps: fixes a :: "atom"   and H :: "chmsg set" shows "(a \<in> atoms H) = ((\<exists>c A B M At. a = At \<and> StatCh c A B (Msg M) \<in> H \<and> At \<in> set M) \<or> (\<exists>c K M At. a = At \<and> DynCh c K (Msg M) \<in> H \<and> At \<in> set M))"
actionI: fixes A :: "'b \<times> 'c \<Rightarrow> bool" assumes "\<And>s t. A (s, t)" shows "\<turnstile> A"
less_gexp_aux_trans: fixes x :: "'a gexp"   and y :: "'a gexp"   and z :: "'a gexp" assumes "local.less_gexp_aux x y"   and "local.less_gexp_aux y z" shows "local.less_gexp_aux x z"
disjoint_list_def: fixes ls :: "'b set list" shows "disjoint_list ls \<equiv> distinct ls \<and> disjoint (set ls)"
IH: shows "fps_of_poly (eulerian_poly k) = Abs_fps (\<lambda>n. of_nat (n + 1) ^ k) * (1 - fps_X) ^ (k + 1)"
config_config_length: fixes A :: "('c list \<Rightarrow> 'd) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> (nat \<times> nat list) \<times> 'd)"   and init :: "'c list"   and qs :: "'c list" shows "length (fst (config A init qs)) = length init"
sets_measure_of_eq: fixes M :: "'c measure"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "sets (measure_of (space M) (sets M) \<mu>) = sets M"
VSagree_def: fixes \<nu> :: "(real, 'b) vec"   and \<nu>' :: "(real, 'b) vec"   and V :: "'b set" shows "VSagree \<nu> \<nu>' V = (\<forall>i\<in>V. \<nu> $ i = \<nu>' $ i)"
R\<^sub>e_def: fixes r :: "rat" shows "R\<^sub>e r = (case quotient_of r of (n, d) \<Rightarrow> floatarith.Num (float_of_int n) / floatarith.Num (float_of_int d))"
A: fixes x :: "'a"   and x' :: "'a"   and a :: "'b" assumes "x \<le> x'" shows "f a x \<le> f' a x'"
partial_equivalence_def: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "partial_equivalence A equiv \<equiv> symmetric A equiv \<and> partial_equivalence_axioms A equiv"
Ints_of_nat: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.of_nat one plus zero n \<in> ring_1.Ints one plus zero minus"
lower_def: shows "lower \<equiv> \<lambda>n. {x. x n \<le> 0}"
vdiff_vinsert_right_nin: fixes a :: "V"   and A :: "V"   and B :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "A -\<^sub>\<circ> vinsert a B = A -\<^sub>\<circ> B"