hcju/isabelleps huggingface dataset:
simps: fixes a1 :: "('a, 'b, 'c, 'd) converter"   and a2 :: "enat" shows "interaction_any_bounded_converter a1 a2 = (\<exists>conv n. a1 = conv \<and> a2 = n \<and> (\<forall>x. interaction_any_bounded_by (run_converter conv x) n) \<and> (\<forall>x xa xb. (xa, xb) \<in> results'_gpv (run_converter conv x) \<longrightarrow> interaction_any_bounded_converter xb n))"
subdegree_le_imp_dvd_left_divring: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "subdegree f \<le> subdegree g" shows "\<exists>k. g = f * k"
tt_ge_iff: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "(v \<preceq>\<^sub>t tt p) = ((p \<noteq> 0 \<or> v = min_term) \<and> (\<forall>u. u \<prec>\<^sub>t v \<longrightarrow> lookup p u = (0::'c)))"
attractive_ordering_axioms: shows "attractive_ordering UNIV (\<le>) (<)"
1: shows "trg\<^sub>D (F f) = map\<^sub>0 (trg\<^sub>C f)"
comp_comp_fun_commute: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "comp_fun_commute f" shows "comp_fun_commute (f \<circ> g)"
toplevel_summands_lderivs_Times: fixes xs :: "'a list"   and r :: "'b rexp"   and s :: "'b rexp" shows "toplevel_summands (lderivs xs (Times r s)) \<subseteq> {Times (lderivs xs r) s} \<union> {r'. \<exists>ys zs. r' \<in> toplevel_summands (lderivs ys s) \<and> ys \<noteq> [] \<and> zs @ ys = xs}"
card_of_refl: fixes A :: "'d set" shows "ordIso2 (card_of A) (card_of A)"
Units_def: shows "left.transformation.Sym = {u \<in> G \<rightarrow>\<^sub>E G. left.transformation.invertible u}"
divide_less_0_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide one a) zero = less a zero"
plus_less: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b" shows "a + c < b + c"
map_insort_wrt: fixes xs :: "'c list"   and r2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and y :: "'c"   and r1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<Longrightarrow> r2 (f y) (f x) = r1 y x" shows "map f (insort_wrt r1 y xs) = insort_wrt r2 (f y) (map f xs)"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (shift_instr instr s1)) \<and> t2 = snd (fst (shift_instr instr s2))"
sum_nonzero_representation_eq: fixes basis :: "'d set"   and v :: "'d" assumes "cindependent basis"   and "v \<in> cspan basis" shows "(\<Sum>b | crepresentation basis v b \<noteq> 0. crepresentation basis v b *\<^sub>C b) = v"
rtrancl_beta_obj_n: fixes L :: "char list set"   and f :: "Label \<Rightarrow> sterm option"   and g :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type" assumes "finite L"   and "\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> ->> t'' \<and> the (g l) = \<sigma>[s,p] t'')"   and "dom f = dom g"   and "\<forall>l\<in>dom f. body (the (f l))" shows "Obj f T ->> Obj g T"
horner_sum_foldr: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'b list" assumes "class.comm_semiring_0 plus zero times" shows "comm_semiring_0.horner_sum plus zero times f a xs = foldr (\<lambda>x b. plus (f x) (times a b)) xs zero"
path_component_of_eq: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "(path_component_of X x = path_component_of X y) = (x \<notin> topspace X \<and> y \<notin> topspace X \<or> x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x y)"
the1_equality': fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<exists>\<^sub>\<le>\<^sub>1 x. P x"   and "P a" shows "(THE x. P x) = a"
is_offending_flows_min_set_minimalize_offending_overapprox: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and ff :: "('c \<times> 'c) list"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph G"   and "SecurityInvariant_withOffendingFlows.is_offending_flows sinvar (set ff) G nP"   and "set ff \<subseteq> edges G"   and "distinct ff" shows "SecurityInvariant_withOffendingFlows.is_offending_flows_min_set sinvar (set (SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox sinvar ff [] G nP)) G nP"
distinct_5: fixes x2 :: "'b" shows "ext0.Val x2 \<noteq> ext0.Top"
one_chain_group: fixes p :: "nat"   and X :: "'b topology" shows "\<one>\<^bsub>chain_group p X\<^esub> = 0"
\<F>_def: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and \<G> :: "'b ltl set"   and j :: "nat" shows "\<F> \<phi> w \<G> j = And (map (\<lambda>i. af\<^sub>G \<phi> (w [i \<rightarrow> j])) [the (threshold \<phi> w \<G>)..<Suc j])"
mupdate_def: fixes v :: "'V"   and wv :: "'W infty"   and \<sigma> :: "('V \<Rightarrow> 'W infty option) \<times> ('V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option)" shows "mupdate v wv \<sigma> \<equiv> ASSERT (update_pre v (\<alpha>s \<sigma>) \<and> wv = mpath_weight' (snd \<sigma> v)) \<bind> (\<lambda>_. let (wl, res) = \<sigma>; pv = mpath' (res v) in FOREACH\<^bsup>uinvarm v (\<alpha>w wl) (\<alpha>r res)\<^esup> (succ ga v) (\<lambda>(w', v') (wl, res). if wv + Num w' < mpath_weight' (res v') then ASSERT (v' \<in> dom wl \<and> pv \<noteq> None) \<bind> (\<lambda>_. ASSERT (wv \<noteq> infty.Infty) \<bind> (\<lambda>_. RETURN (wl(v' \<mapsto> wv + Num w'), res(v' \<mapsto> ((v, w', v') # the pv, val wv + w'))))) else RETURN (wl, res)) (wl, res))"
Swap: fixes B :: "('a, 'b) form"   and A :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<turnstile> B # A # G" shows "\<turnstile> A # B # G"
flat_ord_def: fixes b :: "'f"   and x :: "'f"   and y :: "'f" shows "flat_ord b x y = (x = b \<or> x = y)"
not_prime_unit: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times x one" shows "\<not> normalization_semidom.prime zero times one normalize x"
conc_fun_fail_iff_1: fixes R :: "('b \<times> 'c) set"   and S :: "'c nres" shows "(\<Down> R S = FAIL) = (S = FAIL)"
assms: shows "S = {} \<or> S homeomorphic ball 0 1"
imp_chain_equiv: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and l :: "fol fm list"   and p :: "fol fm" shows "semantics e f g (foldr Imp l p) = ((\<forall>q\<in>set l. semantics e f g q) \<longrightarrow> semantics e f g p)"
assms: shows "x \<in> {s} \<and> x' \<in> {t}"
assms: fixes k :: "nat" shows "\<forall>\<^sub>F n in \<U>. f n \<noteq> k"
normalize_sort_def: fixes sub :: "(String.literal \<times> String.literal) set"   and S :: "String.literal set" shows "normalize_sort sub S = {c \<in> S. \<not> (\<exists>c'\<in>S. class_les sub c' c)}"
iso_blocks_map_inj: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and x :: "'c set"   and y :: "'c set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "x \<in># \<B>"   and "y \<in># \<B>"   and "bij_map ` x = bij_map ` y" shows "x = y"
polysub_def: fixes p :: "'b poly"   and q :: "'b poly" shows "p -\<^sub>p q = p +\<^sub>p ~\<^sub>p q"
nths_vars1': fixes vars1 :: "nat set"   and vars2 :: "nat set"   and dims :: "nat list" assumes "partial_state2 vars1 vars2" shows "nths (partial_state2.dims0 dims vars1 vars2) (partial_state2.vars1' vars1 vars2) = partial_state2.dims1 dims vars1"
nat_less_numeral_power_cancel_iff: fixes a :: "int"   and x :: "num"   and n :: "nat" shows "(nat a < numeral x ^ n) = (a < numeral x ^ n)"
6: shows "monom P (q (deg R q)) (deg R q) \<otimes>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p) = monom P (q (deg R q) \<otimes> p (deg R p)) (deg R p + deg R q)"
less_eq_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "(x \<le> y) = (fst x < fst y \<or> fst x \<le> fst y \<and> snd x \<le> snd y)"
irrefl: fixes a :: "'x \<Rightarrow>\<^sub>0 nat" shows "\<not> a \<prec> a"
assms_3: shows "\<lbrace>p \<cdot> s\<rbrace>x\<lbrace>p\<rbrace>"
bound_intersect_2d_nonempty: fixes prec :: "nat"   and Xs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real"   and m :: "real"   and M :: "real" assumes "bound_intersect_2d prec Xs x = Some (m, M)" shows "(\<Union>(p1, p2)\<in>set Xs. closed_segment p1 p2) \<inter> {p. fst p = x} \<noteq> {}"
simps_1: fixes uu :: "'a" shows "split_bst uu \<langle>\<rangle> = (\<langle>\<rangle>, \<langle>\<rangle>)"
prems: shows "\<forall>y. (y, x) \<in> option_of \<longrightarrow> eq_game (plus_game y (neg_game y)) zero_game"
order_less_subst2: fixes a :: "'k"   and b :: "'k"   and f :: "'k \<Rightarrow> 'l"   and c :: "'l" assumes "a < b"   and "f b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "f a < c"
seq_comp: fixes c1 :: "com"   and s1 :: "char list \<Rightarrow> int"   and s2 :: "char list \<Rightarrow> int"   and c2 :: "com"   and s3 :: "char list \<Rightarrow> int" assumes "(c1, s1) \<rightarrow>* (SKIP, s2)"   and "(c2, s2) \<rightarrow>* (SKIP, s3)" shows "(c1;; c2, s1) \<rightarrow>* (SKIP, s3)"
isometry_cblinfun_compose: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "isometry A"   and "isometry B" shows "isometry (A o\<^sub>C\<^sub>L B)"
numbers_of_marks_simps_3: fixes n :: "nat"   and bs :: "bool list" shows "numbers_of_marks n (False # bs) = numbers_of_marks (Suc n) bs"
that: fixes a :: "'a" assumes "atom a \<sharp> x" shows "thesis"
simps_6: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "rec_option f1 f2 None = f1"
nonempty_Basis: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "Basis \<noteq> {}"
phull_subset_row_space: fixes ps :: "('d \<Rightarrow>\<^sub>0 'e) list"   and ts :: "'d list" assumes "Keys (set ps) \<subseteq> set ts" shows "phull (set ps) \<subseteq> row_to_poly ts ` row_space (polys_to_mat ts ps)"
ext_compat_list_def: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" shows "ext_compat_list ext \<equiv> ext ext \<and> ext_compat_list_axioms ext"
rreq_dip_kD_oip_sqn: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:18..PRreq-:21} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> (osn \<xi> < sqn (rt \<xi>) (oip \<xi>) \<or> sqn (rt \<xi>) (oip \<xi>) = osn \<xi> \<and> the (dhops (rt \<xi>) (oip \<xi>)) \<le> Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (oip \<xi>)) = val))"
Goal: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"   and "x \<sharp> P"   and "c \<sharp> P"   and "c \<noteq> x" shows "c \<sharp> P'"
IH: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and n :: "nat" assumes "pre_subdivFace' g f u v n ovs"   and "minGraphProps g"   and "f \<in> \<F> g"   and "one_final_but g (if u = v then {} else Edges (u # between (vertices f) u v @ [v]))" shows "one_final (subdivFace' g f v n ovs)"
TimesLR_PlusR: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "TimesLR r (Plus s t) = Plus (TimesLR r s) (TimesLR r t)"
all': shows "\<forall>as' a as''. asx = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n'"
complement_unique: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and a :: "'b"   and x :: "'b"   and y :: "'b" assumes "boolean_algebra conj disj compl zero one"   and "conj a x = zero"   and "disj a x = one"   and "conj a y = zero"   and "disj a y = one" shows "x = y"
cut_cf: fixes \<Gamma> :: "'b formula multiset"   and A :: "'b formula"   and \<Delta> :: "'b formula multiset"   and \<Gamma>' :: "'b formula multiset"   and \<Delta>' :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> A, \<Delta>"   and "A, \<Gamma>' \<Rightarrow> \<Delta>'" shows "\<Gamma> + \<Gamma>' \<Rightarrow> \<Delta> + \<Delta>'"
deg_minf_pol_0: fixes p :: "'a" assumes "p \<in> carrier R" shows "(deg R S X p = - \<infinity>) = (p = \<zero>)"
bdd_above_uminus: fixes X :: "'b set" shows "bdd_above (uminus ` X) = bdd_below X"
ax5: fixes v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>[v$ \<noteq> $v]_v"
path_AppendE: fixes l1 :: "'loc"   and l3 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and l2 :: "'loc"   and s :: "'sum"   and l2' :: "'loc" assumes "path l1 l3 (xs @ [(l2, s, l2')])" obtains "l3 = l2'" and "path l1 l2 xs" and "s \<in>\<^sub>A summary l2 l2'"
append_child_def: fixes ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" shows "append_child ptr child = insert_before ptr child None"
execute_while_unfolds_2: fixes a :: "'b array"   and key :: "'b"   and i :: "nat ref"   and h :: "heap"   and f :: "'c Heap" assumes "success (shiftr_p a key i) h"   and "\<not> cond (shiftr_p a key i) h" shows "execute (while (shiftr_p a key i) f) h = execute (return ()) h"
mat_of_cols_list_def: fixes nr :: "nat"   and cs :: "'a list list" shows "Matrix.mat_of_cols_list nr cs = Matrix.mat nr (length cs) (\<lambda>(i, j). cs ! j ! i)"
comp_eq_id_dest: fixes a :: "'c \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'c"   and c :: "'a \<Rightarrow> 'b"   and v :: "'a" assumes "a \<circ> b = id \<circ> c" shows "a (b v) = c v"
ball_Un: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x::'b\<in>(A::'b set) \<union> (B::'b set). (P::'b \<Rightarrow> bool) x) = ((\<forall>x::'b\<in>A. P x) \<and> (\<forall>x::'b\<in>B. P x))"
part_def: fixes s :: "char list \<Rightarrow> int" shows "part s = (\<lambda>v. Some (s v))"
zero_not_in_Basis: shows "0 \<notin> cart_basis"
simps: fixes uu :: "'a \<Rightarrow> 'b list"   and uv :: "'f \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm set list"   and uw :: "'a \<times> 'f \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qAbs set list"   and rhoL :: "('a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm set option) list" shows "envsOfS (ParS uu uv uw rhoL) = set rhoL"
rel_gpv''_mono: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "A \<le> A'"   and "C \<le> C'"   and "R' \<le> R" shows "rel_gpv'' A C R \<le> rel_gpv'' A' C' R'"
QR_lemma_01: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "(\<lambda>x. x mod int q) ` QR.E p q = GAUSS.E q (int p)"
proposition3_4_inv_lists: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list"   and \<beta> :: "'a list"   and \<alpha> :: "'a" assumes "trans r"   and "irrefl r"   and "(r|\<sigma>| -s r \<down>l \<beta>, {#\<alpha>#}) \<in> mul_eq r" shows "\<exists>\<sigma>1 \<sigma>2 \<sigma>3. (\<sigma> = \<sigma>1 @ \<sigma>2 @ \<sigma>3 \<and> set \<sigma>1 \<subseteq> r \<down>l \<beta> \<and> length \<sigma>2 \<le> 1 \<and> set \<sigma>2 \<subseteq> {\<alpha>}) \<and> set \<sigma>3 \<subseteq> r \<down>l (\<alpha> # \<beta>)"
alternating_S_no_flip: fixes s :: "'w"   and t :: "'w"   and n :: "nat" assumes "s \<in> S"   and "t \<in> S"   and "order.greater n 0"   and "order.greater (add_order (s + t)) n \<or> add_order (s + t) = 0" shows "sum_list (alternating_list n s t) \<noteq> sum_list (alternating_list n t s)"
sba1_bot_def: shows "sba1_bot \<equiv> THE x. \<forall>z. x = - (z \<squnion> - z)"
norm_PXtrans: fixes P :: "pol"   and x :: "nat"   and Q :: "pol"   and Q2 :: "pol" assumes "isnorm (PX P x Q)"   and "isnorm Q2" shows "isnorm (PX P x Q2)"
assms: fixes u :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "\<forall>n. u n \<in> s"   and "a \<in> s"   and "u \<longlonglongrightarrow> a" shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
range_oDeriv_subset: fixes A :: "ordinal set" assumes "normal_set A" shows "range (oDeriv (OrdinalVeblen.ordering A)) \<subseteq> A"
le_fin_concat_start: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a list"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind" shows "traces.le_fin ind (w @ w\<^sub>1) (w @ w\<^sub>2) = traces.le_fin ind w\<^sub>1 w\<^sub>2"
nt_mem0: fixes a :: "'a"   and n :: "int" assumes "a \<in> carrier A" shows "n\<triangleright>a\<^bsub>A\<^esub> \<in> carrier A"
C3: fixes y :: "name"   and \<Psi>' :: "'b"   and zvec :: "name list"   and R :: "('a, 'b, 'c) psi" assumes "y \<sharp> \<Psi>'"   and "zvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>*zvec\<rparr>R), \<lparr>\<nu>*zvec\<rparr>\<lparr>\<nu>y\<rparr>R) \<in> Rel"
coeff_eq: fixes p :: "'a mpoly"   and q :: "'a mpoly" shows "(MPoly_Type.coeff p = MPoly_Type.coeff q) = (p = q)"
to_fun_X_plus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "UP_cring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "UP_cring.to_fun R (X_poly_plus R a) b = b \<oplus>\<^bsub>R\<^esub> a"
valid0: shows "valid_msaux args (\<tau> \<sigma> ne) aux0 auxlist0"
star_sim1: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x ; z \<le> z ; y" shows "x\<^sup>\<star> ; z \<le> z ; y\<^sup>\<star>"
assms_2: shows "\<omega> \<in>\<^sub>\<circ> \<alpha>"
extended_ord_pm_powerprod_axioms: shows "extended_ord_pm_powerprod (\<preceq>) (\<prec>)"
msubst_const: fixes P :: "('a, 'b) uexpr"   and v :: "('c, 'b) uexpr" shows "P\<lbrakk>x\<rightarrow>v\<rbrakk> = P"
sets: fixes i :: "nat" shows "(\<lambda>x. x !! i) \<in> N \<rightarrow>\<^sub>M M"
CondAnd: fixes Env :: "env"   and B :: "lname set"   and e1 :: "expr"   and E1 :: "assigned"   and e2 :: "expr"   and E2 :: "assigned"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1"   and "Env\<turnstile> B \<union> assigns_if True e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2"   and "nrm A = B \<union> assigns_if True (BinOp CondAnd e1 e2) \<inter> assigns_if False (BinOp CondAnd e1 e2)"   and "brk A = (\<lambda>l. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>BinOp CondAnd e1 e2\<rangle>\<guillemotright> A"
Lxy_project: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "x \<noteq> y"   and "x \<in> set xs"   and "y \<in> set xs"   and "distinct xs"   and "x < y in xs" shows "Lxy xs {x, y} = [x, y]"
emeasure_bot: fixes X :: "'b set" shows "emeasure bot X = 0"
zero_le_even_power': 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 a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a (2 * n))"
assms_4: fixes i :: "'b" assumes "i \<in> I" shows "space M \<in> F i"
get_scdom_component_owner_document_same: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and thesis :: "bool"   and ptr :: "(_) object_ptr" assumes "\<And>owner_document. \<lbrakk>h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document; cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r owner_document \<in> set sc\<rbrakk> \<Longrightarrow> thesis"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "thesis"
covering_alt_def_all: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "t_covering_design point_set block_collection u_block_size grouping min_index" shows "\<forall>ps\<subseteq>point_set. int (card ps) = grouping \<longrightarrow> min_index \<le> int (block_collection index ps)"
invertible_def: fixes u :: "'a" assumes "u \<in> M" shows "invertible u = (\<exists>v\<in>M. u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
comp_arr_dom: fixes f :: "'s"   and a :: "'s" assumes "S.arr f"   and "S.dom f = a" shows "f \<cdot>\<^sub>S a = f"
zig_zig: fixes r1' :: "'b tree"   and r :: "'b tree"   and p :: "'b"   and r2' :: "'c tree"   and s :: "'b tree"   and a :: "'b"   and u :: "'b tree"   and b :: "'b" assumes "size r1' \<le> size r"   and "real (T_part p r) + \<Phi> r1' + \<Phi> r2' - \<Phi> r \<le> 2 * \<phi> r + 1" shows "real (T_part p r + 1) + \<Phi> \<langle>\<langle>s, a, u\<rangle>, b, r1'\<rangle> + \<Phi> r2' - \<Phi> \<langle>s, a, \<langle>u, b, r\<rangle>\<rangle> \<le> 2 * \<phi> \<langle>s, a, \<langle>u, b, r\<rangle>\<rangle> + 1"
domComm: fixes a :: "'a \<Rightarrow> 'b option"   and b :: "'a \<Rightarrow> 'c option" shows "dom a \<inter> dom b = dom b \<inter> dom a"
hyps_3: fixes Q :: "'state" assumes "\<And>Q'. \<lbrakk>Q \<Rightarrow> Q'; P' \<approx>\<cdot> Q'\<rbrakk> \<Longrightarrow> thesis"   and "P'' \<approx>\<cdot> Q" shows "thesis"
dist_vec_nth_le: fixes x :: "('a, 'b) vec"   and i :: "'b"   and y :: "('a, 'b) vec" shows "dist (x $ i) (y $ i) \<le> dist x y"
adds_triv_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a (plus b a)"
simps_2: fixes n :: "nat"   and m :: "nat"   and \<omega> :: "nat \<Rightarrow> 'a" shows "C n (Suc m) \<omega> = C n m \<omega> \<bind> eP (n + m)"
as_good_asIff: fixes x :: "'a"   and y :: "'a" shows "(x \<in> as_good_as y carrier relation) = (x \<succeq> y \<and> y \<succeq> x)"
poly_order_exists_lemma: fixes p :: "'a list"   and d :: "nat"   and a :: "'a" assumes "length p = d"   and "poly p \<noteq> poly []" shows "\<exists>n q. p = mulexp n [- a, 1::'a] q \<and> poly q a \<noteq> (0::'a)"
dropWhile_last: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<not> P x" shows "last (dropWhile P xs) = last xs"
Collect_mono_iff: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"